Imprimitive Symmetric Graphsresearchers.ms.unimelb.edu.au/~sanming@unimelb/PDF/thesis1.pdf ·...

Imprimitive Symmetric Graphs

Sanming Zhou

This thesis is presented for

the Degree of Doctor of Philosophy ofThe University of Western Australia

Department of Mathematics and Statistics

The University of Western AustraliaPerth, WA 6907

Australia

August 2000

Copyright: c© Sanming Zhou

To My Parents

Abstract

A finite graph Γ is said to be G-symmetric if G is a group of automorphisms of Γ

acting transitively on the ordered pairs of adjacent vertices of Γ. In most cases, the

group G acts imprimitively on the vertices of Γ, that is, the vertex set of Γ admits

a nontrivial G-invariant partition B. The purpose of this thesis is to study such

graphs, called imprimitive G-symmetric graphs.

In the first part of the thesis, we discuss in detail the geometric approach, in-

troduced by Gardiner and Praeger in 1995, for studying imprimitive symmetric

graphs which we use throughout. According to this approach, three configurations

can be associated with (Γ,B), namely the quotient graph ΓB of Γ with respect to

B, the bipartite subgraph Γ[B,C] of Γ induced by two adjacent blocks B,C of B,

and a certain 1-design D(B) induced on B (possibly with repeated blocks). The

approach involves an analysis of these configurations and addresses the problem of

reconstructing Γ from the triple (ΓB,Γ[B,C],D(B)).

In the second part, we study the case where the block size k of D(B) is one less

than the block size v of B. We first assume that D(B) contains no repeated blocks,

and prove that, under the assumption k = v − 1 ≥ 2, this occurs precisely when ΓB

is (G, 2)-arc transitive. In this case, we find a very natural and simple construction

of Γ from ΓB and the induced action of G on B, and prove that up to isomorphism

it produces all such graphs Γ. If in addition ΓB is a complete graph, then we classify

all the possibilities for (Γ, G). We show that Γ[B,C] ∼= Kv−1,v−1 if and only if ΓB

is (G, 3)-arc transitive, and that Γ[B,C] is a matching of v − 1 edges and ΓB is not

a complete graph if and only if ΓB is a certain near n-gonal graph for some even

integer n ≥ 4. In the general case where D(B) may contain repeated blocks, we

give a construction of such graphs from G-point- and G-block-transitive 1-designs,

and prove further that up to isomorphism it gives rise to all such graphs. By using

this, we then classify such graphs arising from the classical projective and affine

geometries.

In the last part, we will investigate the influence of certain “local” actions induced

by the setwise stabilizer GB on the structure of Γ, with emphasis on the case where

Γ is G-locally quasiprimitive. In particular, we will study the case where the actions

of GB on B and on the neighbourhood of B in ΓB are permutationally isomorphic.

i

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Professor Cheryl

E. Praeger, for her careful guidance, continuous encouragement and excellent super-

vision throughout the work on this thesis. It is a tremendous fortune that I have

had an opportunity to do my Ph.D with her. I am most grateful to her for intro-

ducing me to the area of symmetric graphs, for sharing with me her vast experience

in mathematics writing, and for her many invaluable suggestions.

I appreciate greatly Dr. Cai Heng Li who, as my co-supervisor, gave me con-

sistant support and advice and contributed a lot of constructive suggestions during

my study in UWA. His friendliness and many helpful discussions will be always

cherished and most appreciated.

I owe my special and heartfelt thanks to Professor Man-Keung Siu at The Uni-

versity of Hong Kong for his supervision, understanding and long-term spiritual

support. I am very grateful to Dr. A. Gardiner for his valuable discussions, and to

Dr. Tim Penttila for his help in sorting out some references in finite geometry.

I wish to thank sincerely my Master’s supervisor, Professor Yixun Lin, for his

teaching and support over many years, and for introducing me to the fascinating

world of graphs. My thanks also go to Professors Gerard J. Chang, Beifang Chen,

Shigeng Hu, Zhiyuan Huang, Huayi Lin, Guizhen Liu, Jinzhong Mao, Yuyuan Qin,

Jianfang Wang, Li Zhou, Xuding Zhu and Dr. Chunsheng Ma for their help and

support in various ways during my academic pursuit.

I am deeply indebted to my wife Jian-Ying Zhang and my son Han Zhou for their

understanding and for bearing my absence from home for work. They are always

the source of love and support.

I acknowledge the support of an Overseas Postgraduate Research Scholarship

from the Australian Department of Education, Employment and Training and a

University Postgraduate Award from The University of Western Australia.

Finally, I would like to dedicate this thesis with love and best wishes to my

parents who nurtured me during an extremely difficult period. Without their love

and caring I would still be a cowboy in a remote area.

ii

Publications Used in This Thesis

The majority of this thesis is based on certain parts of the following publications.

As a coauthor or the sole author, I was involved actively in the research, planning

and writing of these papers.

1. A. Gardiner, Cheryl E. Praeger and Sanming Zhou, Cross ratio graphs, Proc.

London Math. Soc., to appear. (Reference [46])

2. Cai Heng Li, Cheryl E. Praeger and Sanming Zhou, A class of finite symmetric

graphs with 2-arc transitive quotients, Math. Proc. Cambridge Philos. Soc.

129 (2000), no. 1, 19-34. (Reference [53])

3. Cai Heng Li, Cheryl E. Praeger, Akshay Venkatesh and Sanming Zhou, Finite

locally quasiprimitive graphs, Discrete Math., to appear. (Reference [54])

4. Sanming Zhou, Almost covers of 2-arc transitive graphs, submitted to J. Lon-

don Math. Soc. (Reference [97])

5. Sanming Zhou, Imprimitive symmetric graphs, 3-arc graphs and 1-designs,

Discrete Math., to appear. (Reference [98])

6. Sanming Zhou, Constructing a class of symmetric graphs, submitted to Euro-

pean J. Combinatorics. (Reference [99])

7. Sanming Zhou, Symmetric graphs and flag graphs, preprint. (Reference [100])

8. Sanming Zhou, Classifying a family of symmetric graphs, Bull. Austral. Math.

Soc. 63 (2001), 329-335. (Reference [101])

iii

List of Key Symbols

Groups and geometries

G,H,K Groups

G.H Semidirect product of G by H

H ≤ G H is a subgroup of G

H G H is a normal subgroup of G

NG(H) Normalizer of H in G, where H ≤ G

CoreG(H) Core of H in G, where H ≤ G

soc(G) Socle of G

αG G-orbit containing α

Gα Stabilizer of α in G

G∆ Setwise stabilizer of ∆ in G

G(∆) Pointwise stabilizer of ∆ in G

fixΩ(T ) Fixed point set

Sn Symmetric group of degree n

An Alternating group of degree n

Zn Additive group of integers modulo n

M11,M12,M22,M23,M24 Mathieu groups

PGL(n, q) Projective general linear group

PSL(n, q) Projective special linear group

PΓL(n, q) Semilinear projective group

AGL(n, q) Affine group

AΓL(n, q) Semilinear affine group

GF(q) Finite field with q elements

Sq(q) Set of squares of GF(q)

V (n, q) n-Dimensional linear space over GF(q)

PG(n, q) Projective geometry

AG(n, q) Affine geometry

c(u, w; y, z) Cross-ratio of the 4-tuple (u, w, y, z)

iv

v

Graphs

Γ,Σ,Ξ Graphs

V (Γ) Vertex set of Γ

Arc(Γ) Arc set of Γ

Arcs(Γ) Set of s-arcs of Γ

Γ(α) Neighbourhood of α in Γ

Γ[X] Subgraph of Γ induced by X

Aut(Γ) Full automorphism group of Γ

val(Γ) Valency of (a regular graph) Γ

girth(Γ) Girth of Γ

diam(Γ) Diameter of Γ

Γ Complement of Γ

n · Γ Union of n vertex-disjoint copies of Γ

Kn Complete graph with n vertices

Km,m Complete bipartite graph with m vertices in each part

Knm Complete n-partite graph with m vertices in each part

Cn Cycle with length n

Pn Path with length n

B G-invariant partition of V (Γ), Γ a G-symmetric graph

ΓB Quotient graph of Γ with respect to B

Γ[B,C] Bipartite subgraph induced by adjacent blocks B,C ∈ B

D(B) 1-Design induced on B ∈ B

Γ(B) Union of Γ(α), for α ∈ B

ΓB(B) Neighbourhood of B in ΓB

ΓB(i, B) Set of blocks of B with distance no more than i from B

ΓB(α) Set of blocks of D(B) incident with α, where α ∈ B

v Block size of B

k Block size of D(B)

r Number of blocks in ΓB(α)

b Valency of ΓB

s Valency of Γ[B,C]

G[α] Subgroup of Gα fixing each C ∈ ΓB(α) setwise

vi

(Continued)

G[B] Subgroup of GB fixing setwise each block in ΓB(B)

G[i,B] Subgroup of G fixing setwise each block in ΓB(i, B)

CR(v; x, n) Untwisted cross-ratio graph

TCR(v; x, n) Twisted cross-ratio graph

Ξ(Σ,∆) 3-Arc graph of Σ with respect to ∆

F(D,Θ,Ψ) Flag graph of (a 1-design) D with respect to (Θ,Ψ)

BN Normal partition induced by N

Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Main results and the structure of the thesis . . . . . . . . . . . . . . . 8

2 Notation, definitions and preliminaries 13

2.1 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Partitions, blocks and primitivity . . . . . . . . . . . . . . . . . . . . 17

2.3 Incidence structures and designs . . . . . . . . . . . . . . . . . . . . . 19

2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Imprimitive symmetric graphs: A geometric approach 23

3.1 Symmetric and highly arc-transitive graphs . . . . . . . . . . . . . . . 23

3.2 The geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Refining the given partition . . . . . . . . . . . . . . . . . . . . . . . 34

4 The case k = v − 1: A general analysis 37

4.1 Notation and preliminary results . . . . . . . . . . . . . . . . . . . . 37

4.2 The case where k = 1 and v = 2 . . . . . . . . . . . . . . . . . . . . . 40

4.3 A general discussion: k = v − 1 ≥ 2 . . . . . . . . . . . . . . . . . . . 42

4.4 Analysing an extreme case . . . . . . . . . . . . . . . . . . . . . . . . 46

5 The case k = v − 1 ≥ 2: D(B) contains no repeated blocks 51

5.1 The case D(B) contains no repeated blocks . . . . . . . . . . . . . . . 51

5.2 The 3-arc graph construction . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Three-arc transitive quotient . . . . . . . . . . . . . . . . . . . . . . . 62

vii

viii

6 Three-arc graphs of complete graphs: Classification 65

6.1 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 The 3-transitive subgroups of PΓL(2, v) . . . . . . . . . . . . . . . . . 68

6.3 Definitions of cross-ratio graphs . . . . . . . . . . . . . . . . . . . . . 70

6.4 Characterizing cross-ratio graphs . . . . . . . . . . . . . . . . . . . . 73

6.5 Affine 3-arc graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6 Mathieu graphs, and the classification

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7 Almost covers of two-arc transitive graphs 81

7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.2 Almost covers of complete graphs . . . . . . . . . . . . . . . . . . . . 86

7.3 Almost covers of non-complete graphs . . . . . . . . . . . . . . . . . . 88

7.4 Locally primitive almost covers . . . . . . . . . . . . . . . . . . . . . 92

7.5 Two-arc transitive near-polygonal graphs . . . . . . . . . . . . . . . . 94

8 Flag graphs: A general construction 97

8.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.2 Flag graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.3 Symmetric graphs with k = 1 . . . . . . . . . . . . . . . . . . . . . . 105

9 The case k = v − 1 ≥ 2: Construction 111

9.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

9.3 Coexisting G-flag graphs of doubly transitive designs . . . . . . . . . 117

9.4 Projective flag graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.5 Affine flag graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

10 Local actions 129

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

10.2 G-invariant partitions induced by G[i,B] . . . . . . . . . . . . . . . . . 132

10.3 Two blocks of D(B) incident with either the same or disjoint subsets

of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

10.4 Locally quasiprimitive graphs . . . . . . . . . . . . . . . . . . . . . . 140

ix

11 Local actions: Heritage of the labelling method 145

11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11.2 The labelling technique . . . . . . . . . . . . . . . . . . . . . . . . . . 147

11.3 The 1-design D(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.4 The case where ρ is incidence-preserving . . . . . . . . . . . . . . . . 154

11.5 Reconstruction of Γ, and 3-arc graphs again . . . . . . . . . . . . . . 158

A The Chinese originals of the quotes 169

Chapter 1

Introduction

Things have roots and branches; affairs have scopes and beginnings.

To know what precedes and what follows will lead one near the WAY.

Confucius (551-479 B.C.), The Great Learning

1.1 Introduction

The study of symmetric graphs has long been one of the main themes in Algebraic

Graph Theory. By definition a graph Γ is G-symmetric if Γ admits G as a group of

automorphisms such that G is transitive on the ordered pairs of adjacent vertices

of Γ. Roughly speaking, in most G-symmetric graphs Γ, the group G acts imprimi-

tively on the vertices of Γ, that is, G is transitive on the vertex set V (Γ) of Γ and

V (Γ) admits a nontrivial G-invariant partition B. In this case Γ is said to be an

imprimitive G-symmetric graph.

This thesis is dedicated to a study of imprimitive symmetric graphs, using a

geometric approach which was first introduced by Gardiner and Praeger in 1995 for

locally primitive symmetric graphs. According to this approach, the following three

configurations can be associated with the triple (Γ, G,B) above:

(i) the quotient graph ΓB of Γ with respect to B;

(ii) the bipartite subgraph Γ[B,C] of Γ induced on B ∪ C with isolated vertices

deleted, where B,C are blocks of B adjacent in ΓB; and

2 Introduction

(iii) the 1-design D(B) = (B,ΓB(B), I) induced on a block B ∈ B such that αIC

for α ∈ B and C ∈ ΓB(B) if and only if α is adjacent to some vertex of C,

where ΓB(B) is the neighbourhood of B in ΓB.

The graph Γ is thus “decomposed” into the “product” of these configurations, and

the approach involves an analysis of them. Clearly the triple (ΓB,Γ[B,C],D(B))

reflects the structure of Γ. In some cases, it even determines Γ uniquely (up to

isomorphism), and this happens in particular when Γ[B,C] is a complete bipartite

graph between B and C. In this case Γ is the lexicographic product of ΓB by an

empty graph on |B| vertices. However, in most cases the triple above does not

determine the graph Γ. We will see a simple example of this in Section 4.2, see

Remark 4.2.1. This suggests the following natural question.

Question 1 To what extent does the triple (ΓB,Γ[B,C],D(B)) determine the graph

Γ ?

As is widely recognized, the class of imprimitive symmetric graphs is very large.

Because of this it might be more fruitful to consider some special classes of imprim-

itive symmetric graphs. With respect to this we propose the following problem.

Problem 1 For certain classes of triples (Σ,Π,D), characterize or classify all pos-

sible (Γ, G,B) such that (ΓB,Γ[B,C],D(B)) = (Σ,Π,D).

The effectiveness of the approach relies not only on a thorough understanding of

the three configurations above but also on the feasibility of reconstructing Γ from

the triple (ΓB,Γ[B,C],D(B)). Therefore, refining Question 1, one may naturally

ask the following question.

Question 2 Under what circumstances can we reconstruct the graph Γ from the

triple (ΓB,Γ[B,C],D(B)) ?

The study in this thesis will be more or less centered around these rather general

problems. We will first discuss in detail the approach above for general imprimitive

symmetric graphs. By using this we will then study an interesting and enlightening

case where, for adjacent blocks B,C of B, each part of the bipartition of Γ[B,C]

has size |B| − 1, that is, there exists a unique vertex in B which is not adjacent to

Literature Review 3

any vertex in C. We will give a construction of such graphs Γ and, in particular,

we will show that Γ can be reconstructed from ΓB and the induced action of G on

B. We will also characterize or classify certain subclasses of such graphs. Finally,

we will analyse the induced actions of the setwise stabilizer GB on the block B and

on the neighbourhood ΓB(B) of B in ΓB, and study the influence of these actions

on the structure of the graph Γ. A more detailed introduction to the main results

in this thesis will be given in Section 1.3.

We now leave this discussion for a while and have an excursion to see some

sample results in the world of symmetric graphs.

1.2 Literature review

Let Γ be a finite, undirected graph and let s be a positive integer. An s-arc of Γ

is a sequence of s + 1 vertices of Γ, not necessarily all distinct, such that any two

consecutive vertices are adjacent and any three consecutive vertices are distinct. If Γ

admits a group G of automorphisms such that G is transitive on the s-arcs of Γ, then

Γ is said to be (G, s)-arc transitive. In most cases, such a graph is also G-vertex-

transitive, and we assume this throughout without mentioning explicitly. Under this

assumption the (G, s)-arc transitivity of Γ implies the (G, s− 1)-arc transitivity of

Γ, for s > 1. Usually a 1-arc is called an arc and a (G, 1)-arc transitive graph is

called a G-symmetric graph.

Investigations of symmetric graphs can be found in the literature as early as in

the 1940’s when Tutte [82] proved that, for a G-symmetric cubic graph Γ, the order

of the stabilizer Gα in G of a vertex α is at most 48. Based on this he proved in

the same paper that there is no finite s-arc transitive cubic graph if s > 5. This

fundamental result stimulated greatly the study of symmetric graphs and highly arc-

transitive graphs, and its far-reaching influence in this area can be felt even after

several decades. For example, by refining the ideas used in [82, 83], Sims [77, 78]

generalized this result considerably. He proved in particular that, for a G-symmetric

cubic graph Γ with G primitive on the vertices, the order of Gα is a divisor of 48. In

a series of papers (see [11] and [27]-[31]), Djokovic (partly with Bouwer) extended

Tutte’s work in several directions. In particular he showed [28] that, if Γ has valency

p+1, for a prime p, and if the automorphism group Aut(Γ) of Γ contains a subgroup

4 Introduction

acting regularly on the s-arcs of Γ, then s ≤ 5 or s = 7. (Moreover [29], p must be a

Mersenne prime if p is odd and s ≥ 2.) Almost immediately, Gardiner [37] pointed

out that requiring the existence of such a regular subgroup is a redundancy, and he

proved that the same bound for s is valid if Γ is a graph with valency p+1, p a prime,

such that Aut(Γ) is transitive on the s-arcs but not on the (s + 1)-arcs of Γ. That

the case s = 7 actually occurs was shown by the graph [4] derived from the families

of points and lines on certain quadric surfaces in finite geometries. In [38] Gardiner

proved further that, if Γ is (G, s)-arc but not (G, s+ 1)-arc transitive such that Gα

is doubly primitive on the neighbourhood Γ(α) of α in Γ, and that the pointwise

stabilizer G(Γ(α)∪α) 6= 1, then we have s ≤ 5 or s = 7 as well. By analysing the

stabilizers of adjacent vertices, Goldschmidt [47] obtained an important extension

of Tutte’s result which inspired a lot of subsequent work on groups and geometries.

In particular, he proved that if G is a group of automorphisms of a cubic graph such

that G is transitive on the edges and the stabilizer in G of a vertex is finite, then

the order of the subgroup of G fixing each of two adjacent vertices divides 27. A

significantly simplified proof of this result was given by Weiss in [94]. In the 1970’s

and early 1980’s, Weiss [88, 89, 90, 91, 92] obtained a number of results regarding

the structure of the stabilizer of a vertex for a finite s-arc transitive (not necessarily

cubic) graph, s ≥ 1.

Also inspired by Tutte’s fundamental result above, a lot of work has been done

in constructing symmetric graphs and highly arc-transitive graphs. Tutte himself

gave the first example of a connected 5-arc transitive cubic graph, and Conway con-

structed (but did not publish, see [6, pp.145]) infinitely many such graphs as covers

of a given one. After that, a number of new 5-arc transitive cubic graphs were

constructed [7, 8, 9, 17]. In [18], Conder found infinitely many new 5-arc transitive

cubic graphs by showing that, for all but finitely many positive integers n, both the

alternating and symmetric groups of degree n may be represented as full automor-

phism groups of 5-arc transitive cubic graphs. In [61], all cubic symmetric graphs

with small girth (up to 6) were determined. An infinite family of 4-arc transitive

cubic graphs each with girth 12 was constructed in [19], and a classification of 4-

and 5-arc transitive cubic graphs with girth less than or equal to 13 was given (with

some exceptions) in [62]. In [57] Lorimer determined all cubic symmetric graphs of

order (that is, the number of vertices) at most 120 which are neither bipartite graphs

Literature Review 5

nor Cayley graphs. A complete classification of cubic symmetric graphs with order

at most 240 was recently given in [22] by using the result (see e.g. [21]) that the

group of a cubic symmetric graph is a homomorphic image of one of seven finitely

presented groups. There are also a few constructions and characterizations of sym-

metric graphs concerning the valency: Lorimer [56, 59] studied symmetric graphs

with prime valency, and Praeger and Xu characterized [73] connected symmetric

graphs of twice prime valency whose automorphism groups have abelian normal p-

subgroups which are not semiregular on vertices. This latter work motivated the

investigation of symmetric graphs of valency 4 conducted in [41, 42] by Gardiner

and Praeger.

From a group-theoretic point of view, a symmetric graph can be defined as

an orbital graph of a transitive permutation group (see e.g. [70, Section 2]). In

[76] Sabidussi introduced a way of identifying the self-paired orbital involved, and

developed a group-theoretic method for constructing an isomorphic copy of the given

symmetric graph (see also [56, 58, 60]). More precisely, a graph is G-symmetric if

and only if it is isomorphic to a certain kind of “coset graph” with vertices the right

cosets in G of a certain subgroup of G. This group-theoretic approach has proved

to be very useful in constructing and classifying some classes of symmetric graphs.

It also indicates the strong connection between groups and symmetric graphs. In

particular the classification of finite simple groups has had a great impact on research

into symmetric graphs (see e.g. [13, 65]). A number of important results have been

proved by using this powerful mathematical tool. The first one of them is the

celebrated theorem of Weiss [93] which asserts that, apart from the cycles, there are

no s-arc transitive graphs for s > 7. As mentioned earlier, 7-arc transitive graphs

do exist; and Conder and Walker [20] prove recently that there are infinitely many

such graphs. In fact, they proved that, for all but finitely many positive integers

n, there are two connected graphs which admit, respectively, the alternating and

symmetric groups of degree n as 7-arc transitive groups of automorphisms. Before

the classification of finite simple groups, Chao [15] classified symmetric graphs with

prime order. By using the classification of finite simple groups, Cheng and Oxley

[16] determined all symmetric graphs with twice prime orders, and Wang and Xu

[87] classified all symmetric graphs with triple prime orders. In [72], Praeger, Wang

and Xu classified all symmetric graphs of order a product of two distinct primes by

6 Introduction

using the classification [74] of all vertex-primitive graphs of order a product of two

distinct primes. All G-symmetric graphs with order 6p such that p ≥ 5 is a prime

and G is solvable were classified in [85].

Naturally, a G-symmetric graph Γ can be called a primitive or imprimitive G-

symmetric graph according to whether G is primitive or imprimitive on the vertices

of Γ. By using the result of [77], Wong [96] determined all primitive cubic symmetric

graphs. As a consequence of the work of Wang [86], which relies on the classification

of finite simple groups, all primitive symmetric graphs of valency 4 are known. In

general, for studying primitive symmetric graphs we need to understand the possible

structures of finite primitive groups. The information needed is contained in the

O’Nan-Scott Theorem (see [55] or [80]), which categorizes finite primitive groups

into several types. This theorem has been proved to be very useful in studying

finite primitive groups and their applications, and in particular in studying primitive

symmetric graphs. Similar to the primitive case, a G-symmetric graph Γ is said

to be quasiprimitive if G is quasiprimitive on the vertices of Γ. (A permutation

group is quasiprimitive if each of its nontrivial normal subgroups is transitive. Any

primitive group is quasiprimitive, but not conversely.) Considering the local action,

a G-symmetric graph Γ is said to be G-locally primitive (G-locally quasiprimitive,

respectively) if in its induced action Gα is primitive (quasiprimitive, respectively)

on Γ(α). Since a G-vertex-transitive graph Γ is (G, 2)-arc transitive if and only if

Gα is 2-transitive on Γ(α) and since 2-transitive groups are primitive, it is clear

that any (G, 2)-arc transitive graph is G-locally primitive, and in turn any G-locally

primitive graph is G-locally quasiprimitive.

As a result of the classification of finite simple groups, all the finite 2-transitive

groups are known (see e.g. [13, 51]). Because of this, an extensive study of 2-arc

transitive graphs has been conducted during the past two decades. It was proved

in [14] that, under certain conditions, a 2-arc transitive graph must be the inci-

dence graph of a (known) symmetric design. In [49], Ivanov investigated 2- but not

3-arc transitive graphs. In [50], Ivanov and Praeger classified all primitive affine

2-arc transitive graphs and all bi-primitive affine 2-arc transitive graphs. (A 2-arc

transitive graph Γ is said to be affine if there is a vector space N and a subgroup

G ≤ Aut(Γ) such that N ≤ G ≤ AGL(N) with N acting regularly on the ver-

tices of Γ and G acting 2-arc transitively on Γ, where AGL(N) is the group of

Literature Review 7

all affine transformations of N and N is identified with the subgroup of transla-

tions.) In [33, 34] Fang and Praeger constructed and classified some classes of 2-arc

transitive graphs admitting a Suzuki group or a Ree group (see also Fang’s PhD

Thesis [35]). Recently, Hassani, Nochefranca and Praeger [48] studied 2-arc transi-

tive graphs admitting a two-dimensional projective linear group. Examples of 2-arc

transitive graphs of girth 5 containing Petersen subgraphs were constructed in [64]

via a certain kind of flag-transitive geometry. A construction is given in [3] for all

the pairs (Γ, G) such that Γ is (G, 2)-arc transitive and G has a minimal normal

subgroup which is nonabelian and regular on the vertices of Γ. A classification of

2-arc transitive circulants was given in [1]. In [68] Praeger gave an O’Nan-Scott

type Theorem for finite quasiprimitive groups, and this has been the impetus for a

lot of work on quasiprimitive symmetric graphs and locally quasiprimitive graphs

conducted by Praeger and her colleagues (see for example [52, 54, 71]). In [68]

Praeger also proved that every finite, non-bipartite, 2-arc transitive graph is a cover

of a quasiprimitive 2-arc transitive graph; and moreover among the possible types

of quasiprimitive groups only four of them (namely, affine type, almost simple type,

product type, twisted wreath type) can appear as a quasiprimitive, 2-arc transitive

group of automorphisms of a connected graph. For bipartite 2-arc transitive graphs,

a useful reduction was given in [69], also by Praeger.

Imprimitive symmetric graphs have been studied in various ways in the literature.

Classical examples of such graphs include the “covering graphs” constructed in [6,

Chapter 19] and some highly arc-transitive graphs constructed in [7, 8, 17, 18, 21, 62].

There have been a few characterizations of some special classes of imprimitive sym-

metric graphs [36, 41, 42, 73, 81]. Nevertheless, unlike the primitive case, it seems

that there is no powerful mathematical tool available for dealing with imprimitive

symmetric graphs. In this sense the main difficulty in studying symmetric graphs

lies in the imprimitive case. Recently, Gardiner and Praeger [43] proposed a geomet-

ric approach to studying imprimitive symmetric graphs and discussed in detail the

case where Γ is G-locally primitive. Further [44, 45], they indicated an extension of

their approach for the whole class of imprimitive symmetric graphs. Recall that a

G-symmetric graph Γ is imprimitive if and only if its vertex set admits a nontrivial

G-invariant partition B. So in this case we have a natural quotient graph ΓB of

Γ with respect to B. According to the approach of Gardiner and Praeger, such a

8 Introduction

graph Γ can be “decomposed” into the “product” of this quotient and two other

configurations, namely the “inter-block” bipartite graph Γ[B,C] and the 1-design

D(B) which we defined in the previous section. It was suggested [43, 44, 45] that

these three configurations may have a strong influence on the structure of Γ.

1.3 Main results and the structure of the thesis

This thesis can be divided into the following three parts.

Part I. The first part consists of this introductory chapter and the next two

chapters. In Chapter 2, we will introduce notation, terminology and preliminary

results for permutation groups, designs and graphs that will be used throughout. In

Chapter 3, we will discuss in detail the geometric approach of Gardiner and Praeger

[43] for studying imprimitive symmetric graphs, and thus set the framework for

the whole thesis. As mentioned in Section 1.1, the triple (ΓB,Γ[B,C],D(B)) will

be associated with any G-symmetric graph Γ admitting a nontrivial G-invariant

partition B. Here Γ[B,C] is the induced bipartite subgraph of Γ with bipartition

Γ(C)∩B,Γ(B)∩C, where Γ(B) is the set of vertices of Γ adjacent to at least one

vertex of B. In most cases we will identify D(B) = (B,ΓB(B), I) with the 1-design

with point set B and blocks Γ(C)∩B (with possible repetition), so D(B) has block

size k = |Γ(C)∩B|, where C runs over all the blocks in the neighbourhood ΓB(B) of

B in ΓB. The study of Γ will involve a detailed analysis of these three configurations,

as well as addressing the problem of the reconstruction of Γ from the triple above.

Part II. The heart of the thesis is the second part, which contains Chapters

4 through 9. Since the class of imprimitive symmetric graphs is very large, it is

unrealistic to discuss all the cases in the thesis. In this part we concentrate on

the case where the block size k of the 1-design D(B) is one less than the block

size v of the partition B. As we will see later, this case is rather enlightening and

unexpectedly rich in both theory and examples. Chapter 4 is devoted to a general

analysis of this case and thus provides a basis for subsequent study in this part. As

fundamental properties for this case, we will prove that the induced action of G on

B is faithful and the induced action of GB on B is 2-transitive, where GB is the

setwise stabilizer of B in G. We will also study (Section 4.4) an extreme case for

which all of Γ, ΓB and Γ[B,C] can be determined explicitly. (Chapter 4 is based on

Main Results 9

certain parts of [53] and [101].)

In Chapter 5, we study the case where in addition D(B) contains no repeated

blocks. Not only is this a natural assumption geometrically, but also we will prove

that, under the assumption k = v− 1 ≥ 2, D(B) contains no repeated blocks if and

only if the quotient ΓB is (G, 2)-arc transitive (Theorem 5.1.2). In this case, the

valency of ΓB is equal to v, the vertices of Γ can be labelled in a natural way by the

arcs of ΓB, and each vertex of Γ has a unique mate. We will give (Section 5.2) a

very natural and simple construction of a class of graphs Γ such that k = v− 1 ≥ 2

and D(B) contains no repeated blocks. Moreover, we will show that every graph

Γ satisfying these conditions can be constructed by using this construction (see

Theorem 5.2.3). The construction bears some similarity to the “covering graph”

construction of Biggs [6, pp.149-154]. The ingredients for our construction are a

(G, 2)-arc transitive graph Σ and a self-paired G-orbit ∆ on 3-arcs of Σ. Given

these, we define the 3-arc graph of Σ with respect to ∆ to be the graph with vertices

the arcs of Σ in which two vertices represented respectively by arcs (σ, τ), (σ′, τ ′) of

Σ are adjacent if and only if (τ, σ, σ′, τ ′) is a 3-arc in ∆. The possibilities for Γ[B,C]

depend on the pair (ΓB, G), and vice versa. For example, under the assumptions

above, we will prove (Theorem 5.3.1) that the extreme case Γ[B,C] ∼= Kv−1,v−1

(where Γ[B,C] contains the maximum possible number of edges) occurs if and only

if ΓB is (G, 3)-arc transitive. (This chapter is based on publication [53].)

The 3-arc graph construction above enables us to classify in Chapter 6 all the

pairs (Γ, G) with Γ a G-symmetric graph such that k = v−1 ≥ 2, D(B) contains no

repeated blocks and ΓB is a complete graph. From this construction the classification

of such graphs Γ is equivalent to classifying all 3-arc graphs of (G, 2)-arc transitive

complete graphs Σ of valency v. In this case G is 3-transitive on the vertices of

Σ. Thus the classification of such (Γ, G) relies on the classification of 3-transitive

permutation groups, and hence depends on the classification of finite simple groups.

The examples of such graphs Γ arising from 3-transitive projective groups are the so-

called cross-ratio graphs, which can be defined in terms of cross ratios of quadruples

of points of the projective line PG(1, v). Other examples of such graphs Γ include two

graphs arising from each of the 3-transitive affine groups, and two graphs arising from

each of the Mathieu groups M11 (degree 12) and M22 (degree 22). The classification

of all possible (Γ, G) will be given in Theorem 6.6.1. (This chapter is based on

10 Introduction

certain parts of [46] and [99].)

Continuing our discussion for the case where k = v − 1 ≥ 2 and D(B) contains

no repeated blocks, we will study in Chapter 7 the second extreme case for Γ[B,C],

namely the case where Γ[B,C] ∼= (v − 1) · K2 is a matching of v − 1 edges (and

thus Γ[B,C] contains the minimum possible number of edges). In this case, we call

Γ an almost cover of ΓB. If in addition ΓB is a complete graph, then using the

result in Chapter 6 we get a classification (Theorem 7.2.1) of all the possibilities for

such (Γ, G). In the general case where ΓB is connected but not complete, we find

a surprising connection between such graphs Γ and an interesting class of graphs,

namely near-polygonal graphs, which are associated with the Buekenhout geometries

[12, 75] of the following diagram:

s ns

cs

More precisely, in this case we will prove (Theorem 7.3.1) that, for some even integer

n ≥ 4, ΓB must be a near n-gonal graph with respect to a G-orbit on n-cycles of ΓB;

and moreover we will show that any (G, 2)-arc transitive near n-gonal graph (where

n is even) with respect to a G-orbit on n-cycles can occur as such a quotient ΓB. (A

near n-gonal graph [75] is a connected graph Σ of girth at least four together with

a set E of n-cycles of Σ such that each 2-arc of Σ is contained in a unique member

of E . In this case we also say that Σ is a near n-gonal graph with respect to E .)

It was known in [43] that any G-locally primitive graph Γ admitting a G-invariant

partition B with k = v− 1 ≥ 2 is an almost cover of ΓB. So the results above apply

in particular to this case, and we get an amended form (see Corollary 7.4.1) of [43,

Theorem 5.4]. We conclude Chapter 7 by giving necessary and sufficient conditions

for a (G, 2)-arc transitive graph of girth at least four to be near-polygonal. In view

of the results above, these are needed in constructing almost covers of (G, 2)-arc

transitive graphs. (This chapter is based on paper [97].)

In Chapter 8 we study a large class of symmetric graphs, namely the class of G-

symmetric graphs such that the dual 1-design of D(B) contains no repeated blocks.

Our study in this chapter reveals a very close connection between such graphs and

certain point- and block-transitive 1-designs. More precisely, we will give a construc-

tion of such graphs Γ from some G-point-transitive and G-block-transitive 1-designs

D, and prove that, up to isomorphism, it produces all of them (see Theorem 8.2.1).

Main Results 11

Each of the constructed graphs, called the G-flag graphs of D, has vertex set a

certain G-orbit Θ on the flags of D which satisfies some natural conditions. In

particular, any G-symmetric graph Γ with k = 1 satisfies the condition above and

thus is isomorphic to a G-flag graph. We will characterize such a graph Γ as a

G-flag graph with Θ satisfying some additional condition (see Theorem 8.3.1). This

chapter is preparatory for the next chapter, and this is the reason why we include

it in the second part. However, we should point out that this chapter is of interest

for its own sake, and the construction above seems to be useful in classifying or

characterizing some interesting classes of symmetric graphs. (This chapter is based

on paper [100].)

In Chapter 9 we return to the general case of k = v−1 ≥ 2 without assuming the

non-repetition of the blocks of D(B). Based on the similar idea as in the previous

chapter, we will give a construction of such graphs Γ from some G-point-transitive

and G-block-transitive 1-designs D, and prove that, up to isomorphism, it produces

all such graphs (see Theorem 9.2.1). In the particular case where the design D

involved is the trivial design with block size 2, the construction gives rise to the

3-arc graphs introduced in Chapter 5. (However, the 3-arc graph construction is

interesting and useful for its own sake.) In the case where D is a certain G-doubly

transitive design, the constructed graph Γ has complete quotient ΓB. Using this

construction we will classify (Theorems 9.4.1 and 9.5.1) all the G-symmetric graphs

Γ such that k = v − 1 ≥ 2 and ΓB is complete when the design D involved is either

the projective geometry PG(n, q) or the affine geometry AG(n, q). (This chapter is

based on [99] and part of [46].)

Part III. The third part of the thesis consists of the last two chapters. The

main purpose of this part is to investigate certain local actions induced by GB, and

to study their influence on the structure of Γ. In Chapter 10 we will study the

induced actions of GB on B and ΓB(B). We will see that the relationships between

the kernels of these two actions affect significantly the structure of Γ, especially in

the case where Γ is G-locally quasiprimitive. In the last chapter, Chapter 11, we will

study a specific case where the actions of GB on B and ΓB(B) are permutationally

equivalent. Geometrically, this requires that the group of automorphisms of D(B)

induced by GB acts in the same way on the points and blocks of D(B). In this case

D(B) plays a more active role in influencing the structure of Γ and ΓB, and we will

12 Introduction

show that a labelling method similar to that used in Chapter 5 applies. Based on

this labelling we will prove among other things that Γ is such a graph if and only if

it is isomorphic to a 3-arc graph of some G-symmetric but not necessarily (G, 2)-arc

transitive graph. (Chapter 11 is based on publication [98].)

Chapter 2

Notation, definitions andpreliminaries

If names are not rectified, then language will not be in accord with

truth; if language is not in accord with truth, then things cannot be

accomplished.

Confucius (551-479 B.C.), Lun Yu [The Analects] 13:3

This chapter is a collection of basic definitions and preliminary results relating

to permutation groups, incidence structures, designs and graphs that will be used

in subsequent chapters.

2.1 Permutation groups

A bijection from a finite set Ω to itself is said to be a permutation of Ω, and the

symmetric group on Ω, denoted by Sym(Ω) as usual, is the group of all permutations

of Ω equipped with the ordinary composition of mappings. Any subgroup G of

Sym(Ω) is said to be a permutation group on Ω. If the size |Ω| of Ω is equal to n,

then we say that G is a permutation group of degree n.

Let G be a finite group. Suppose that, for each α ∈ Ω and g ∈ G, there

corresponds a member of Ω, denoted by αg. We say that this correspondence defines

an action of G on Ω, or G acts on Ω, if for any α ∈ Ω and g, h ∈ G the following

(i)-(ii) hold:

(i) α1 = α, where 1 is the identity of the group G;

14 Notation, definitions and preliminaries

(ii) (αg)h = αgh.

In other words, an action of G on Ω is a mapping (α, g) 7→ αg from Ω × G to Ω

which satisfies the conditions (i), (ii) above. In such a case, the degree of the action

of G on Ω is defined to be |Ω|. We say that an element g of G fixes a point α of

Ω if αg = α. The kernel of the action of G on Ω is defined to be the subgroup of

all elements of G which fix each point of Ω. If this kernel is equal to the identity

subgroup of G, then G is said to act faithfully on Ω.

Example 2.1.1 Every permutation group G on Ω acts naturally on Ω, where αg is

the image of α under g, for α ∈ Ω and g ∈ G. Clearly, such an action of G on Ω

is faithful. Except where stated otherwise, we will always assume that this is the

action we are dealing with whenever we have a permutation group.

Closely related to group actions is the concept of permutation representation.

By definition a permutation representation of a group G on a finite set Ω is a group

homomorphism ϕ : G → Sym(Ω). For such a permutation representation ϕ, the

image (G)ϕ of G is a permutation group on Ω, denoted by GΩ. Thus, G acts on Ω via

the natural action of (G)ϕ on Ω. That is, for α ∈ Ω and g ∈ G we define αg := α(g)ϕ,

the image of α under the permutation (g)ϕ of Ω. It is clear that the kernel of

this action of G on Ω is exactly the kernel Ker(ϕ) of the group homomorphism ϕ,

and if Ker(ϕ) = 1 then we say that the permutation representation ϕ is faithful.

Conversely, if G acts on a finite set Ω, then each g ∈ G induces a permutation g of

Ω defined by g : α 7→ αg for α ∈ Ω. Hence such an action determines a permutation

representation ϕ of G on Ω, defined by ϕ : g 7→ g for g ∈ G, whose kernel is exactly

the kernel of the action of G on Ω.

Example 2.1.2 (Right multiplication) Suppose G is a group and H ≤ G is a sub-

group of G. Let [G : H ] be the set of right cosets of H in G. Then (Ha)g = Hag,

for Ha ∈ [G : H ] and g ∈ G, defines an action of G on [G : H ]. One can see that the

kernel of this action is⋂

g∈GHg, which is called the core of H in G and is denoted

by CoreG(H), where Hg := g−1Hg.

Naturally, an action of G on Ω induces an equivalence relation ∼G on Ω defined

by

Permutation Groups 15

α ∼G β if and only if αg = β for some g ∈ G.

The equivalence classes of ∼G are said to be G-orbits on Ω. So any two G-orbits are

either identical or disjoint, and the G-orbit containing a given point α of Ω is

αG := αg : g ∈ G.

We say that G is transitive on Ω if there is only one G-orbit on Ω, and that G is

intransitive on Ω otherwise. For a subset ∆ of Ω, we define

∆g := αg : α ∈ ∆.

In particular, if ∆g = ∆ for each g ∈ G, then ∆ is said to be G-invariant; in this

case G induces an action on ∆. We call the subgroups

G∆ := g ∈ G : ∆g = ∆

and

G(∆) := g ∈ G : αg = α for each α ∈ ∆

the setwise stabilizer and the pointwise stabilizer of ∆ in G, respectively. In partic-

ular, for α, β, γ ∈ Ω, the subgroup Gα := Gα of G is called the stabilizer of α in

G, and we set Gαβ := (Gα)β, Gαβγ := (Gαβ)γ, etc. If Gα acts trivially on Ω, that

is, Gα = CoreG(Gα) for all α ∈ Ω, then G is said to be semiregular on Ω. If G

acts transitively and semiregularly on Ω, then we say that G is regular on Ω. The

following results can be found in standard books on permutation groups (see e.g.

[26, 95]).

Lemma 2.1.1 Suppose that G is a group acting on a finite set Ω. Let α ∈ Ω and

g, h ∈ G. Then

(a) Gαg = g−1Gαg.

(b) αg = αh if and only if Gαg = Gαh.

(c) |αG| · |Gα| = |G|. In particular, G is transitive on Ω if and only if |Ω| =

|G : Gα|; and in this case G acts regularly on Ω if and only if |Ω| = |G : CoreG(Gα)|.

Now suppose G1, G2 are groups acting on finite sets Ω1,Ω2, respectively. If there

exist a bijection ρ : Ω1 → Ω2 and a group homomorphism ψ : G1 → G2 such that

ρ(αg) = (ρ(α))ψ(g)


for all α ∈ Ω1 and g ∈ G1, then the action of G1 on Ω1 is said to be permutationally

isomorphic to the action of G2 on Ω2. In particular, if a group G acts on both Ω1 and

Ω2, then we say that the actions of G on Ω1 and Ω2 are permutationally equivalent

if there exists a bijection ρ : Ω1 → Ω2 such that

ρ(αg) = (ρ(α))g

for all α ∈ Ω1 and g ∈ G.

For a positive integer k, we use Ω(k) to denote the set of k-tuples of distinct

members of Ω. Let G act on Ω. Then G induces a natural action on Ω(k) defined by

(α1, α2, . . . , αk)g := (αg1, α

g2, . . . , α

gk)

for (α1, α2, . . . , αk) ∈ Ω(k) and g ∈ G. If, under this action, G is transitive on Ω(k),

then G is said to be k-transitive on Ω; if G is regular on Ω(k), then G is said to be

sharply k-transitive on Ω. As a consequence of the finite simple group classification,

all 2-transitive permutation groups are known up to permutation isomorphism. The

following classification theorem is from [13]. For a group G, the socle soc(G) of G

is defined to be the product of all minimal normal subgroups of G.

Theorem 2.1.1 ([13, pp.8]) Let G be a finite 2-transitive permutation group of de-

gree n. Then either soc(G) is an elementary abelian group, or soc(G) is a nonabelian

simple group and one of the cases listed in Table 1 (next page) occurs, where k is

the maximum degree of transitivity of the group G.

As mentioned in [13], the socle of G is k-transitive in all cases in Table 1 except

where (i) G = Sn is n-transitive while soc(G) = An is (n − 2)-transitive, (ii) G ≤

PΓL(2, q) (with q odd) is 3-transitive while soc(G) = PSL(2, q) is 2-transitive, and

(iii) G = Ree(3) is 2-transitive of degree 28 while soc(G) = PSL(2, 8) is 1-transitive.

Note that the 2-transitive groups with abelian socles are not included in Table 1.

For such groups, one can consult [51]. In particular, any group G with AGL(d, q) ≤

G ≤ AΓL(d, q) is 2-transitive on the d-dimensional vector space over GF(q); and

moreover G is 3-transitive if and only if G = AGL(d, 2), or G = Z42.A7 < AGL(4, 2).

Partitions, Blocks and Primitivity 17

soc(G) n k RemarksAn, n ≥ 5 n n Two representations if n = 6PSL(d, q), d ≥ 2 (qd − 1)/(q − 1) 3 if d = 2 (d, q) 6= (2, 2), (2, 3)

2 if d > 2 Two representations if d > 2PSU(3, q) q3 + 1 2 q > 2Suz(q) q2 + 1 2 q = 22a+1 > 2Ree(q) q3 + 1 2 q = 32a+1 > 3PSp(2d, 2) 22d−1 + 2d−1 2 d > 2PSp(2d, 2) 22d−1 − 2d−1 2 d > 2PSL(2, 11) 11 2 Two representationsPSL(2, 8) 28 2A7 15 2 Two representationsM11 11 4M11 12 3M12 12 5 Two representationsM22 22 3M23 23 4M24 24 5HS 176 2 Two representationsCo3 276 2

Table 1 Socles of 2-transitive groups

Finally, an element of a finite group G is called a 2-element if its order in G is a

power of 2. In particular, an element of G with order 2 is called an involution of G.

We refer to [26, 95] for terminology and notation on permutation groups not defined

above.

2.2 Partitions, blocks and primitivity

A partition of a finite set Ω is a set B of subsets of Ω such that⋃

B∈B B = Ω and

B ∩ C = ∅ for distinct B,C ∈ B. We call each member of B a block of B, and if B

has n blocks then we say that it is an n-partition of Ω. Clearly, α : α ∈ Ω and

Ω are partitions of Ω, which we call the trivial partitions of Ω. For two partitions

B1,B2 of Ω, we say that B1 is a refinement of B2 if each block of B2 is a union of

some blocks of B1; and we say that B1 is a genuine refinement of B2 if in addition

B1 6= α : α ∈ Ω and B1 6= B2.

Let B be a partition of Ω and let G act on Ω. If Bg ∈ B for any B ∈ B and

g ∈ G, then B is said to be a G-invariant partition of Ω. In such a case, G permutes


blockwise the blocks of B and thus induces a natural (possibly unfaithful) action

on B. Obviously, the trivial partitions α : α ∈ Ω, Ω of Ω are G-invariant.

Suppose G is transitive on Ω. If the trivial partitions are the only G-invariant

partitions of Ω, then G is said to be primitive on Ω; otherwise G is said to be

imprimitive on Ω. In general, if G is k-transitive on Ω, for some k > 1, such that

the pointwise stabilizer in G of any k − 1 distinct points of Ω is primitive on the

remaining points, then G is said to be k-primitive on Ω.

Note that each block B of a G-invariant partition of Ω is a block of imprimitivity

for G in Ω in the sense that, for each g ∈ G, either Bg = B or Bg ∩ B = ∅.

Conversely, for a transitive group G acting on Ω, any block B of imprimitivity for G

in Ω induces a G-invariant partition of Ω, namely Bg : g ∈ G; and in this case each

block of this partition is also a block of imprimitivity for G in Ω. Thus, a partition B

of Ω is G-invariant if and only if each block of B is a block of imprimitivity for G in

Ω. Hence G is primitive on Ω if and only if the only blocks of imprimitivity for G in

Ω are Ω and α, for α ∈ Ω. Clearly, we have Gα ≤ GB ≤ G for α ∈ B. Conversely,

for any subgroup H of G with Gα ≤ H ≤ G, the H-orbit B := αH containing α is a

block of imprimitivity for G in Ω, and hence B induces a G-invariant partition of Ω.

Further, if Gα ≤ H1 ≤ H2 < G, then αH1 ⊆ αH2 and the partition corresponding to

H1 refines the partition corresponding to H2. So the lattice of G-invariant partitions

of Ω (with partial order the refinement of partitions) is isomorphic to the lattice of

subgroups H of G containing Gα. Therefore, G is primitive on Ω if and only if Gα

is a maximal subgroup of G.

In the following we will write GB,C := (GB)C , GB,C,D := (GB,C)D for B,C,D ∈

B. For any subset T of G, we define

fixΩ(T ) := α ∈ Ω : αg = α for all g ∈ T,

the fixed point set of T in Ω. The following lemma will be used in our later discussion.

Lemma 2.2.1 ([26, pp.19]) If a group G acts transitively on a finite set Ω, then,

for each α ∈ Ω, fixΩ(Gα) is a block of imprimitivity for G in Ω.

We conclude this section by giving the definition of quasiprimitivity, which relies

on the following result.

Incidence Structures and Designs 19

Lemma 2.2.2 (see e.g. [70, Lemma 10.1]) Let a group G act on a finite set Ω, and

let N be a normal subgroup of G. Then the set of N-orbits on Ω is a G-invariant

partition of Ω.

We will denote this partition by BN and called it the G-normal partition of Ω

induced by N . Clearly, the trivial partitions α : α ∈ Ω and Ω are G-normal

partitions induced by the identity subgroup and G itself, respectively. If these are

the only G-normal partitions of Ω, then G is said to be quasiprimitive on Ω. In other

words, G is quasiprimitive on Ω if and only if every non-indentity normal subgroup

of G is transitive on Ω. Thus, G is quasiprimitive on Ω implies in particular that

it is transitive on Ω. It follows from the definition that G is primitive on Ω implies

that it is quasiprimitive on Ω. Note that the converse of this is not true (see e.g. [70,

Section 10]).

2.3 Incidence structures and designs

We refer to [5, 10] for terminology and notation on design theory. An incidence

structure is a triple D = (V,B, I), where V , B are disjoint finite sets and I is a

binary relation between V and B, that is, I ⊆ V × B. The members of V , B and

I are called the points, blocks and flags of D, respectively. If (α,X) is a flag of D,

then we simply write αIX and say that α,X are incident with each other. The trace

of a block X (a point α, respectively) of D is the subset α ∈ V : αIX of V (the

subset X ∈ B : αIX of B, respectively). If two blocks have the same trace, then

they are said to be repeated blocks of D. As usual in the literature, in the case where

D contains no repeated blocks we may identify each block with its trace and thus

identify B with a set of subsets of V . If the traces of all blocks of D have the same

cardinality k (which we call the block size of D) and if the traces of all points of D

have the same cardinality r, then D is said to be a 1-(v, k, r) design, where v := |V |.

In such a case, we have vr = bk by counting the number of flags of D in two different

ways, where we set b := |B|. A 1-(v, k, r) design D is said to be a t-(v, k, λ) design,

for some integers t ≥ 2 and λ ≥ 1, if any t distinct points are incident with λ blocks

simultaneously. The dual of a 1-(v, k, r) design D = (V,B, I) is the 1-(b, r, k) design

D∗ := (B, V, I∗) with XI∗α if and only if αIX. For two 1-designs D = (V,B, I) and

D′ = (V ′,B′, I′), an isomorphism from D to D′ is a bijection ψ : V ∪ B → V ′ ∪ B′


such that ψ(V ) = V ′, ψ(B) = B′, and αIX if and only if ψ(α)I′ψ(X). If there exists

an isomorphism ψ from D to D∗, then D is said to be self-dual; and if moreover

ψ2 = 1 then ψ is called a polarity of D. An isomorphism from D to itself is said to be

an automorphism of D, and all such automorphisms form the (full) automorphism

group of D, denoted by Aut(D). In general, if G is a group acting on the points and

the blocks of D respectively such that the incidence relation of D is preserved by

these actions, that is, αIX if and only if αgIXg for α ∈ V , X ∈ B and g ∈ G, then

we say that D admits G as a group of automorphisms. In this case G induces an

action on the flags of D. If G is transitive on the points (blocks, flags, respectively)

of D, then D is said to be G-point-transitive (G-block-transitive, G-flag-transitive,

respectively). As a convention, when we say D is G-transitive, we mean it is G-

point-transitive. Similar convention applies to G-doubly transitive 1-designs. For a

point α of D, we set Bα := X \ α : X ∈ B, αIX and let Iα be the incidence

relation between V \ α and Bα induced by I. If D is a 2-(v, k, λ) design, then

Dα := (V \ α,Bα, Iα) is a 1-(v − 1, k − 1, λ− 1) design, and in this case D is said

to be an extension of Dα.

A linear space [5] is an incidence structure of points and blocks (called lines) in

which any two distinct points are incident with exactly one line, any point is incident

with at least two lines, and any line with at least two points. A linear space with

each line incident with exactly two points is called a trivial linear space.

2.4 Graphs

All the graphs in this thesis will refer to finite, undirected and simple graphs. Such

a graph Γ can be defined as an incidence structure (V,E, I) with no repeated blocks

such that each block is incident with exactly 2 points. The members of V,E are

called the vertices and edges of Γ, respectively. As usual, we use V (Γ) and E(Γ)

to denote respectively the vertex set V and the edge set E of Γ, and thus we write

Γ = (V (Γ), E(Γ)). Two vertices α, β of Γ are said to be adjacent if there exists an

edge e of Γ which is incident with both α and β. In such a case, we say that e

joins α and β and we may identify e with the unordered pair α, β. So we may

identify E(Γ) with the set of all such unordered pairs of vertices of Γ. For α ∈ V (Γ),

we use Γ(α) to denote the neighbourhood of α in Γ, that is, the set of vertices of Γ

Graphs 21

adjacent to α. The valency of α in Γ is defined to be the size of Γ(α). A vertex of

Γ with valency 0 is called an isolated vertex of Γ. If all vertices of Γ have the same

valency, then Γ is said to be regular. In this case, this common valency is called

the valency of Γ and is denoted by val(Γ). A complete graph is a graph in which

any two distinct vertices are adjacent, whilst an empty graph is a graph in which

any two vertices are not adjacent. A subgraph of Γ is a graph Σ = (V (Σ), E(Σ))

with V (Σ) ⊆ V (Γ), E(Σ) ⊆ E(Γ). For a subset X of V (Γ), we use Γ[X] to denote

the subgraph of Γ induced by X, that is, the graph with vertex set X in which

α, β ∈ X are adjacent if and only if they are adjacent in Γ. In particular, if Γ[X] is

a complete graph, then X is said to be a clique of Γ; and if Γ[X] is an empty graph,

then X is said to be an independent set of Γ. The graph Γ is said to be an n-partite

graph if V (Γ) admits an n-partition with each block an independent set of Γ. If in

addition any two vertices in distinct parts of this n-partition are adjacent, then Γ

is said to be a complete n-partite graph. In particular, a 2-partite graph is called a

bipartite graph. For two graphs Γ = (V (Γ), E(Γ)) and Σ = (V (Σ), E(Σ)) (with or

without common vertices), the union of Γ and Σ, denoted by Γ ∪ Σ, is the graph

with vertex set V (Γ)∪ V (Σ) and edge set E(Γ)∪E(Σ). The union of finitely many

graphs is defined similarly. In particular, we will use n · Γ to denote the union of n

vertex-disjoint copies of Γ. The lexicographic product Γ[Σ] of Γ by Σ is defined to

be the graph with vertex set V (Γ)× V (Σ) in which (α, β) and (σ, τ) are adjacent if

and only if either α, σ are adjacent in Γ, or α = σ and β, τ are adjacent in Σ. We

will use Γ to denote the complement of a graph Γ, that is, the graph with the same

vertices as Γ in which α, β are adjacent if and only if they are not adjacent in Γ.

A path of a graph Γ of length n is a sequence α0, α1, . . . , αn of n + 1 distinct

vertices such that αi−1, αi are adjacent for i = 1, 2, . . . , n. Such a path is said to

connect α0 and αn. Define a binary relation ∼Γ on V (Γ) such that α ∼Γ β if and

only if there exists a path of Γ connecting α and β. Then it is an equivalence relation

on V (Γ), and we call the subgraphs of Γ induced by the equivalence classes of ∼Γ

the connected components of Γ. The graph Γ is said to be connected if it has only one

connected component, and disconnected otherwise. The distance in Γ between two

given vertices α, β, denoted by dΓ(α, β) (or simply d(α, β) if no ambiguity exists),

is the shortest length of a path of Γ connecting α and β if they are in the same

connected component of Γ, and is defined to be ∞ otherwise. (As a convention, we


define d(α, α) = 0.) The diameter of Γ, denoted by diam(Γ), is the largest distance

in Γ between any two vertices of Γ. For n ≥ 3, an n-cycle (or a cycle of length n) of

Γ is an (n+1)-tuple (α0, α1, . . . , αn−1, α0) of vertices of Γ such that α0, α1, . . . , αn−1

are pairwise distinct and αi−1, αi are adjacent for i = 1, 2, . . . , n (subscripts modulo

n). The girth of Γ, denoted by girth(Γ), is the shortest length of a cycle of Γ if Γ

contains cycles, and is defined to be ∞ otherwise.

Let Γ, Σ be two graphs. A mapping ψ : V (Γ) → V (Σ) is called a (graph)

homomorphism from Γ to Σ if ψ maps adjacent vertices of Γ to adjacent vertices of

Σ. If in addition ψ is one-to-one, then it is called a (graph) monomorphism; and if

in addition ψ is a bijection with ψ−1 a homomorphism from Σ to Γ, then ψ is said

to be a (graph) isomorphism from Γ to Σ. In particular, an isomorphism from Γ

to itself is said to be an automorphism of Γ. All such automorphisms of Γ form a

subgroup of Sym(V (Γ)), called the full automorphism group of Γ and denoted by

Aut(Γ). Any subgroup of Aut(Γ) is called an automorphism group of Γ. In general,

if G is a group acting on V (Γ) such that, for any g ∈ G, two vertices α, β of Γ are

adjacent in Γ implies that αg, βg are adjacent in Γ, then we say that Γ admits G as

a group of automorphisms.

We will use Kn, Pn, Cn, Km,m, Knm to denote, respectively, the complete graph on

n vertices, the path of length n, the cycle of length n, the complete bipartite graph

with m vertices in each part of its bipartition, and the complete n-partite graph

with m vertices in each part of its n-partition. The graph n · K2 with n edges is

called a matching.

Chapter 3

Imprimitive symmetric graphs: Ageometric approach

From TAO proceeds the one; one produces two; this makes three.

From these three proceed all things. All things thus bear the imprint of

the negative yin behind and embrace the positive yang in front, and

through the blending of the vital force (ch’i) they achieve harmony.

Lao Tzu (6th or 4th Cent. B.C. ?), Tao Te Ching 42

The geometric approach we will use in this thesis was first introduced by Gardiner

and Praeger in [43]. Although their paper was written in the context of G-locally

primitive graphs, the same approach is well suitable for studying general imprimitive

G-symmetric graphs, and the theory was extended to such graphs in [44, 45, 53]. In

this chapter we will introduce this approach and prove some basic results involved,

and thus set the framework for the whole thesis. Most results in Section 3.2 were

known explicitly or implicitly in [43, 44, 45, 66]. We start with some definitions

relating to symmetric graphs.

3.1 Symmetric and highly arc-transitive graphs

Let Γ be a graph and s a positive integer. An s-arc of Γ is an (s + 1)-tuple

(α0, α1, . . . , αs) of vertices of Γ such that αi−1 is adjacent to αi for 1 ≤ i ≤ s

and αi−1 6= αi+1 for 1 ≤ i ≤ s−1. We will use Arcs(Γ) to denote the set of s-arcs of

Γ. Usually a 1-arc is called an arc, and instead of Arc1(Γ) we use Arc(Γ) to denote

the set of arcs of Γ.

24 Geometric approach

Suppose that Γ admits a group G as a group of automorphisms. Then, in

a natural way, G induces an action on Arcs(Γ) defined by (α0, α1, . . . , αs)g :=

(αg0, αg1, . . . , α

gs), for (α0, α1, . . . , αs) ∈ Arcs(Γ) and g ∈ G. If, under this induced

action, G is transitive on Arcs(Γ), then Γ is said to be (G, s)-arc transitive. As

usual in the literature, a (G, 1)-arc transitive graph is called a G-symmetric graph,

or simply a symmetric graph if the group G is not important in the context. Like-

wise, if G is transitive on V (Γ), then Γ is said to be G-vertex-transitive. Clearly,

any G-vertex-transitive graph is regular.

Perhaps it is the right time to say a few words about the definition of a (G, s)-arc

transitive graph. In most cases (but not always), such a graph Γ is also G-vertex-

transitive; and this is the case researchers are interested in. In fact, if Γ is a G-

symmetric graph with no isolated vertices, then it must be G-vertex-transitive since

each vertex of Γ can be taken as the initial vertex of an arc. In general, it follows

from [70, Theorem 9.3] that, if Γ is a (G, s)-arc transitive graph with each connected

component containing at least one s-arc, then either Γ is G-vertex-transitive and is

(G, i)-arc transitive for each i with 1 ≤ i ≤ s, or the connected components of Γ are

isomorphic trees. Therefore, as usual in the literature, we will be concerned with

G-vertex-transitive, (G, s)-arc transitive graphs only.

Convention 3.1.1 By a (G, s)-arc transitive (G-symmetric, respectively) graph,

we will always refer to a G-vertex-transtive, (G, s)-arc transitive (G-symmetric,

respectively) graph with valency at least one.

As we see above, under the assumption of G-vertex-transitivity, (G, s)-arc transi-

tivity implies (G, s−1)-arc transitivity. Here, for s = 1, we may interprete (G, 0)-arc

transitivity as G-vertex-transitivity. Conversely, if Γ is (G, s−1)-arc transitive and,

for some fixed (s− 1)-arc (α0, α1, . . . , αs−1) of Γ, the stabilizer Gα0α1...αs−1 is transi-

tive on Γ(αs−1) \ αs−2, then Γ is (G, s)-arc transitive. In particular we have part

(b) of the following lemma. Part (a) of this lemma follows from the definition of a

G-symmetric graph.

Lemma 3.1.1 Let Γ be a G-vertex-transitive graph, and let α ∈ V (Γ). Then the

following (a)-(b) hold.

(a) Γ is G-symmetric if and only if Gα is transitive on Γ(α).

(b) Γ is (G, 2)-arc transitive if and only if Gα is 2-transitive on Γ(α).

Geometric Approach 25

We should warn that, for s ≥ 3, the similar assertion “Γ is (G, s)-arc transitive

if and only if Gα is s-transitive on Γ(α)” is not valid. In view of (a) above, if Γ

is G-symmetric such that Gα is primitive on Γ(α), then Γ is said to be G-locally

primitive. Similarly, if Γ is G-symmetric such that Gα is quasiprimitive on Γ(α),

then Γ is said to be G-locally quasiprimitive. In general, for a given property P,

if the action of Gα on Γ(α) has the property P, then following [66] we say that Γ

is G-locally P. Since any 2-transitive group is primitive, it follows from (b) above

that any (G, 2)-arc transitive graph is G-locally primitive. Similarly, any G-locally

primitive graph is G-locally quasiprimitive.

Since the objects studied in this thesis are imprimitive symmetric graphs, we

now give a formal definition for such graphs.

Definition 3.1.1 Suppose Γ is a G-symmetric graph. If G acts imprimitively on

V (Γ), then Γ is said to be an imprimitive G-symmetric graph.

Finally, if a graph Γ admits G as a group of automorphisms, then G induces a

natural action on the edges of Γ defined by α, βg := αg, βg, for α, β ∈ E(Γ)

and g ∈ G. If, under this action, G is transitive on E(Γ), then Γ is said to be

G-edge-transitive.

3.2 The geometric approach

Suppose Γ is an imprimitive G-symmetric graph. Then it follows from the definition

that V (Γ) admits a nontrivial G-invariant partition B. For a vertex α of Γ, we will

always use B(α) to denote the (unique) block of B containing α. Since B is G-

invariant, we have

B(αg) = (B(α))g (3.1)

for any α ∈ V (Γ) and g ∈ G. A standard approach to studying such a graph Γ is to

analyse the quotient graph of Γ with respect to B, denoted by ΓB, which is defined

to be the graph with vertex set B in which two blocks B,C ∈ B are adjacent if and

only if there exists at least one edge of Γ joining a vertex of B and a vertex of C. To

extract useful information about Γ from this quotient graph, we require naturally

that ΓB is a nonempty graph. In this case we have the following lemma (see [6,

Proposition 22.1] or [66, Lemma 1.1(c)] for the “only if” part).


Lemma 3.2.1 Suppose that Γ is a nonempty G-symmetric graph whose vertex set

admits a nontrivial G-invariant partition B. Then ΓB is a nonempty graph if and

only if each block of B is an independent set of Γ.

Proof Since Γ is nonempty, there exists an arc (α, β) of Γ. The G-symmetry of

Γ implies that each arc of Γ has the form (αg, βg) for some g ∈ G. So from (3.1)

we have: ΓB is a nonempty graph ⇔ B(α) 6= B(β) for some arc (α, β) of Γ ⇔

B(αg) 6= B(βg) for all arcs (αg, βg) of Γ (where g ∈ G) ⇔ each block of B is an

independent set of Γ. 2

In other words, ΓB is an empty graph if and only if each block of B consists of

connected components of Γ. In order to avoid this somewhat trivial case, we make

the following convention throughout this thesis.

Convention 3.2.1 For the pair (Γ,B) above, we always assume that ΓB is a non-

empty graph. Thus each block of B is an independent set of Γ.

The following lemma shows that ΓB inherits the G-symmetry from Γ.

Lemma 3.2.2 ([66, Lemma 1.1(a)]) Suppose that Γ is a G-symmetric graph and B

is a nontrivial G-invariant partition of V (Γ). Then ΓB is G-symmetric under the

induced action of G on B.

Proof Since G is transitive on V (Γ) and B is a G-invariant partition of V (Γ), it

follows thatG is transitive on B, that is, ΓB isG-vertex-transitive. Let (B,C), (D,E)

be two arcs of ΓB. Then there exist α ∈ B, β ∈ C, γ ∈ D, δ ∈ E such that

(α, β), (γ, δ) ∈ Arc(Γ). By the G-symmetry of Γ, there exists g ∈ G such that

(α, β)g = (γ, δ). From (3.1) above, this implies Bg = (B(α))g = B(αg) = B(γ) = D,

and similarly Cg = E. Hence (B,C)g = (D,E) and ΓB is G-symmetric. 2

We remark that, if Γ is connected, then ΓB is connected as well ([66, Lemma

1.1(b)]). Since the connected components of a symmetric graph are all symmetric

and are pairwise isomorphic, without loss of generality we may even require that

ΓB is connected. Nevertheless we will not assume this in most parts of this thesis.

(Whenever we need the connectedness of ΓB we will state this explicitly.)


The induced action of G on B is not necessarily faithful. If the kernel of this

action is K, then BKg := Bg, for B ∈ B and Kg ∈ G/K, defines a faithful action of

G/K on B. Moreover, under this action ΓB is (G/K)-symmetric.

The quotient graph ΓB conveys a lot of information about the graph Γ. Never-

theless, it does not determine Γ completely since it does not tell us how adjacent

blocks of B are joined by edges of Γ. To compensate for this shortage, we need to

consider the “inter-block” subgraph Γ[B,C] of Γ induced by (Γ(C)∩B)∪(Γ(B)∩C),

where B,C are adjacent blocks of B, where for any block D ∈ B we set

Γ(D) :=⋃

α∈D

Γ(α).

Since each block of B is an independent set of Γ, this subgraph Γ[B,C] is a bipartite

graph with bipartition Γ(C)∩B,Γ(B)∩C. Denote by ΓB(B) the neighbourhood

of B in ΓB. To depict genuinely the structure of Γ we also need a “cross-sectional”

geometry, namely the incidence structure D(B) := (B,ΓB(B), I) in which a point

α ∈ B and a block C ∈ ΓB(B) are incident if and only if α is adjacent in Γ to at

least one vertex of C. Clearly, the trace of the block C of D(B) is Γ(C) ∩ B. We

denote by ΓB(α) the trace of α in D(B), that is,

ΓB(α) := C ∈ ΓB(B) : α ∈ Γ(C).

We will show in the following that, up to isomorphism, Γ[B,C] and D(B) are re-

spectively independent of the choice of adjacent blocks B,C and the block B, and

that D(B) is in fact a 1-design (see Lemmas 3.2.3(a) and 3.2.5(a) below). Thus,

with any imprimitive G-symmetric graph Γ and nontrivial G-invariant partition B

of V (Γ) we have associated three configurations, namely the quotient graph ΓB, the

bipartite graph Γ[B,C], and the 1-design D(B). In a very informal way we can say

that the graph Γ is “decomposed” into the “product” of these three configurations

which, according to Gardiner and Praeger [43], might have a strong influence on the

structure of Γ. The usefulness of this geometric approach to studying imprimitive

symmetric graphs lies in the following two aspects:

(i) a detailed analysis of the three configurations above; and

(ii) an attempt at reconstructing Γ from the triple (ΓB,Γ[B,C],D(B)).


In the subsequent chapters we will follow this approach and study some specific

classes of imprimitive symmetric graphs.

Now let us prove some basic properties regarding Γ[B,C] and D(B).

Lemma 3.2.3 Suppose the triple (Γ, G,B) is as in Lemma 3.2.2. Then the following

(a)-(c) hold for adjacent blocks B,C of B.

(a) The bipartite graph Γ[B,C] is, up to isomorphism, independent of the choice

of adjacent blocks B,C of B.

(b) ([66, Lemma 1.4(b)]) Γ[B,C] is (GB∪C)-symmetric and (GB,C)-edge-transitive.

(c) Γ(C) ∩ B and Γ(B) ∩ C are two (GB,C)-orbits on V (Γ).

(d) |GB∪C : GB,C | = 2, and hence GB,C GB∪C .

Proof Let B,C;D,E be two pairs of adjacent blocks of B. Then (B,C), (D,E) are

arcs of ΓB. Hence by Lemma 3.2.2 there exists g ∈ G such that (B,C)g = (D,E).

The restriction g of g on B∪C is a bijection from B∪C to D∪E. Since g preserves

the adjacency of Γ, g is an isomorphism form Γ[B,C] to Γ[D,E], and thus (a) is

proved.

Now let us prove (b). Since Γ is G-symmetric, for any two arcs (α, β), (γ, δ) of

Γ[B,C], there exists g ∈ G such that (α, β)g = (γ, δ). Since either α, γ are in the

same block of B,C and β, δ in the other, or α, δ are in the same block of B,C and β, γ

in the other, it follows from (3.1) that (B,C)g = (B,C) or (C,B) respectively. So

g ∈ GB∪C and hence Γ[B,C] is (GB∪C)-symmetric. For any two edges α, β, γ, δ

of Γ[B,C], we may assume without loss of generality that α, γ ∈ B and β, δ ∈ C.

Then there exists h ∈ G such that (α, β)h = (γ, δ). Again by (3.1) we have h ∈ GB,C .

Thus Γ[B,C] is (GB,C)-edge-transitive, and hence (b) is proved. Since each vertex

in Γ(C) ∩ B is incident with an edge of Γ[B,C] and since GB,C is transitive on the

edges of Γ[B,C], we conclude that GB,C is transitive on Γ(C) ∩ B. Similarly, GB,C

is transitive on Γ(B) ∩C. Since Γ(C) ∩B and Γ(B) ∩C are (GB,C)-invariant, part

(c) follows.

Finally, for any g, h ∈ GB∪C \ GB,C and x ∈ GB,C , one can check that g−1xh ∈

GB,C and hence g−1(GB,C)h = GB,C . Therefore, |GB∪C : GB,C | = 2 and thus

GB,C GB∪C . 2


Lemma 3.2.4 Suppose the triple (Γ, G,B) is as in Lemma 3.2.2, and let α ∈ V (Γ).

Then the following (a)-(b) hold.

(a) ([66, Lemma 1.4(a)]) Γ(α) admits a Gα-invariant partition, namely Γ(α) ∩

C : C ∈ ΓB(α). Moreover, Gα is transitive on the blocks of this partition.

(b) If Γ[B,C] is a matching, then the blocks of this partition are singletons and

the actions of Gα on Γ(α) and ΓB(α) are permutationally equivalent.

Proof Set P := Γ(α) ∩ C : C ∈ ΓB(α). Then clearly P is a partition of Γ(α).

For any block Γ(α) ∩ C of P and g ∈ Gα, one can easily check that (Γ(α) ∩ C)g =

Γ(α) ∩ Cg ∈ P and hence P is Gα-invariant. Let Γ(α) ∩D be a second block of P.

Then there exist β ∈ C, γ ∈ D which are adjacent to α. So there exists h ∈ G such

that (α, β)h = (α, γ). Thus h ∈ Gα and (3.1) implies Ch = D. Therefore, we have

(Γ(α)∩C)h = Γ(α)∩Ch = Γ(α)∩D, which implies that Gα is transitive on P and

hence (a) is proved.

If Γ[B,C] is a matching for adjacent blocks B,C of B, then (a) implies that

each C ∈ ΓB(α) contains a unique vertex adjacent to α and hence ρ : β 7→ B(β),

for β ∈ Γ(α), defines a bijection from Γ(α) to ΓB(α). From (3.1), we then have

ρ(βg) = B(βg) = (B(β))g = (ρ(β))g for g ∈ Gα, and hence (b) follows. 2

From Lemma 3.2.3(a), we know that

k := |Γ(B) ∩ C|

is independent of the choice of adjacent blocks B,C of B. Since G is transitive on

V (Γ), the value

r := |ΓB(α)|

is independent of the choice of α ∈ V (Γ). Consequently, the incidence structure

D(B) is a 1-(v, k, r) design, where

v := |B|

denotes the block size of B. Similarly, the G-vertex-transitivity of Γ and Lemma

3.2.4(a) together imply that

s := |Γ(α) ∩ C|


is independent of the choice of the flag (α,C) of D(B) (where α ∈ B and C ∈ ΓB(α)).

Clearly, we have val(Γ) = rs and val(Γ[B,C]) = s. Let

b := val(ΓB)

denote the valency of ΓB. Then b is equal to the number of blocks of D(B). Through-

out this thesis we will reserve these symbols v, r, b, k, s for the above-defined param-

eters with respect to the G-invariant partition B of V (Γ). In particular, if k = v,

s = 1, then the bipartite graph Γ[B,C] is a perfect matching between B and C, and

in this case the graph Γ is called a cover of ΓB. In general, if k = v, then following

[54], Γ is said to be a multicover of ΓB. Similarly, if k = v − 1 and s = 1, that is,

Γ[B,C] ∼= (v − 1) · K2, then we say that Γ is an almost cover of ΓB and that ΓB

is almost covered by Γ. Lemma 3.2.4 has the following consequence for G-locally

primitive graphs.

Corollary 3.2.1 ([43, Lemma 3.1(b)]) Suppose that Γ is a G-locally primitive graph

and B is a nontrivial G-invariant partition of V (Γ). Then either

(a) Γ[B,C] ∼= k ·K2 is a matching of k edges, for adjacent blocks B,C of B; or

(b) Γ is a bipartite graph with each part of the bipartition of a connected compo-

nent contained in some block of B, the traces of any two distinct blocks of D(B) are

disjoint (thus D(B) contains no repeated blocks), and v = bk.

Proof Let B ∈ B and α ∈ B. By Lemma 3.2.4(a), Γ(α) ∩ C : C ∈ ΓB(α) is

a Gα-invariant partition of Γ(α). Since Γ is G-locally primitive, for C ∈ ΓB(α) we

have either (i) |Γ(α) ∩ C| = 1; or (ii) Γ(α) ∩ C = Γ(α). In the first case, we have

Γ[B,C] ∼= k · K2 and (a) occurs. In the second case, we have Γ(α) ⊆ C. This,

together with Lemma 3.2.3, implies that the connected component of Γ containing

α is a bipartite graph with one part of its bipartition contained in Γ(C)∩B and the

other contained in Γ(B)∩C. Thus Γ is a bipartite graph. Moreover, we have r = 1

and the traces of any two blocks of D(B) are disjoint. Hence v = bk and (b) occurs.

2

Lemma 3.2.5 Suppose the triple (Γ, G,B) is as in Lemma 3.2.2. Then the following

(a)-(c) hold for B ∈ B.


(a) D(B) is a 1-(v, k, r) design; moreover, up to isomorphism, it is independent

of the choice of B ∈ B.

(b) GB induces a group of automorphisms of D(B) which is transitive on the

points, the blocks and the flags of D(B).

(c) G induces a transitive action on the set of triples (D(B), α, C), where B ∈ B

and (α,C) is a flag of D(B).

Proof (a) That D(B) is a 1-(v, k, r) design has been shown earlier. For two blocks

B,D ∈ B, there exists g ∈ G such that Bg = D. So we have (ΓB(B))g = ΓB(D)

and hence g induces a bijection from B ∪ ΓB(B) to D ∪ ΓB(D). For α ∈ B and

C ∈ ΓB(B), we have: (α,C) is a flag of D(B) ⇔ α ∈ Γ(C) ∩B ⇔ αg ∈ Γ(Cg) ∩Bg

⇔ αg ∈ Γ(Cg)∩D⇔ (αg, Cg) is a flag of D(D). Therefore, g induces an isomorphism

from D(B) to D(D).

(b) Similarly, each g ∈ GB induces a bijection from B∪ΓB(B) to itself such that

Bg = B and (ΓB(B))g = ΓB(B). We have: (α,C) is a flag of D(B) ⇔ α ∈ Γ(C)∩B

⇔ αg ∈ Γ(Cg) ∩ B ⇔ (αg, Cg) is a flag of D(B). So g induces an automorphism of

D(B). Therefore, GB induces a group of automorphisms of D(B).

Since Γ is G-vertex-transitive, GB is transitive on the point set B of D(B).

Since ΓB is G-symmetric (Lemma 3.2.2), by Lemma 3.1.1(a) GB is transitive on the

block set ΓB(B) of D(B). Suppose (α,C), (β,D) are two flags of D(B). Then α

is adjacent to a vertex γ ∈ C and β is adjacent to a vertex δ ∈ D. By the G-

symmetry of Γ, there exists g ∈ G such that (α, γ)g = (β, δ). This implies g ∈ GB

and (α,C)g = (β,D), and hence GB is transitive on the flags of D(B).

(c) Clearly, (D(B), α, C)g := (D(Bg), αg, Cg), for g ∈ G, defines an action of G

on the set of triples (D(B), α, C) such that B ∈ B and (α,C) is a flag of D(B).

Since G is transitive on B and GB is transitive on the flags of D(B), as shown in

(b), we see that G is transitive on the set of such triples. 2

By Lemma 3.2.5(b), the number of times a block C of D(B) is repeated is

independent of the choice of B,C. We call this number the multiplicity of D(B).

For most of the time, we will view D(B) as the 1-(v, k, r) design with point set B

and blocks the subsets Γ(C)∩B of B (for C ∈ ΓB(B)) each repeated m times, where

m is the multiplicity of D(B).


We conclude this section by examining certain “local actions” induced by the

action of G on V (Γ). For B ∈ B, the pointwise stabilizer G(B) is, by definition,

the kernel of the action of GB on B. We will use G[B] to denote the kernel of the

action of GB on ΓB(B). Thus, G[B] is the subgroup of GB fixing each C ∈ ΓB(B)

blockwise. For α ∈ V (Γ), the stabilizer Gα of α in G induces a natural action on

ΓB(α). We denote by G[α] the kernel of this action, that is, G[α] is the subgroup of

Gα fixing each B ∈ ΓB(α) setwise. One can see that G[α] is also the subgroup of Gα

fixing each Γ(α) ∩B setwise, and hence it induces a natural action on Γ(α) ∩B. A

detailed study of the influence of these “local actions” on the structure of Γ will be

conducted in Chapters 10 and 11. For the moment we content ourselves with the

following lemma the proof of which is straightforward. For a vertex α ∈ V (Γ) and

a block B ∈ B, we set Gα,B := (Gα)B.

Lemma 3.2.6 Suppose the triple (Γ, G,B) is as in Lemma 3.2.2.

(a) For B,C ∈ B, the following (i)-(ii) hold.

(i) The action of GB on B and the action of GC on C are permutationally

isomorphic; and the action of GB on ΓB(B) and the action of GC on

ΓB(C) are permutationally isomorphic.

(ii) The action of G(B) on ΓB(B) and the action of G(C) on ΓB(C) are per-

mutationally isomorphic.

(b) For any α, β ∈ V (Γ), the following (i)-(iii) hold.

(i) The action of G(B(α)) on Γ(α) and the action of G(B(β)) on Γ(β) are

permutationally isomorphic. In particular, if α, β ∈ B, then the actions

of G(B) on Γ(α) and Γ(β) are permutationally isomorphic.

(ii) The action of G(B(α)) on ΓB(α) and the action of G(B(β)) on ΓB(β) are

permutationally isomorphic. In particular, if α, β ∈ B, then the actions

of G(B) on ΓB(α) and ΓB(β) are permutationally isomorphic.

(iii) The action of Gα on ΓB(α) and the action of Gβ on ΓB(β) are permuta-

tionally isomorphic.

(c) For any α ∈ V (Γ) and B,C ∈ ΓB(α), the following (i)-(ii) hold.


(i) The action of Gα,B on Γ(α) ∩B and the action of Gα,C on Γ(α) ∩C are

permutationally isomorphic.

(ii) The actions of G[α] on Γ(α) ∩ B and on Γ(α) ∩ C are permutationally

isomorphic.

Proof (a) Since G is transitive on B, there exists g ∈ G such that Bg = C. So g

induces a bijection ρ fromB to C. By Lemma 2.1.1(a), x 7→ g−1xg for x ∈ GB defines

an isomorphism from GB to GC . Since, for γ ∈ B, ρ(γx) = (γx)g = (γg)g−1xg =

(ρ(γ))g−1xg, the first assertion in (i) follows. We have (ΓB(B))g = ΓB(C) and g

induces a bijection λ : D 7→ Dg from ΓB(B) to ΓB(C). Since λ(Dx) = (Dx)g =

(Dg)g−1xg = (λ(D))g

−1xg, the second assertion in (i) then follows.

Similarly, for any x ∈ G(B), we have g−1xg ∈ G(C) and x 7→ g−1xg defines an

isomorphism from G(B) to G(C). One can see that the action of G(B) on ΓB(B) is

permutationally isomorphic to the action of G(C) on ΓB(C) with respect to λ.

(b) Since G is transitive on V (Γ), there exists g ∈ G such that αg = β. Hence

g induces a bijection ρ : Γ(α) → Γ(β) defined by ρ : γ 7→ γg for γ ∈ Γ(α). For

each x ∈ G(B(α)), one can see that g−1xg ∈ G(B(β)) and hence x 7→ g−1xg defines an

isomorphism from G(B(α)) to G(B(β)). It is clear that, for any γ ∈ Γ(α) and x ∈ G(B),

we have ρ(γx) = (γx)g = (γg)g−1xg = (ρ(γ))g

−1xg, and hence (i) follows. Similarly,

g induces a bijection λ : ΓB(α) → ΓB(β) defined by λ : D 7→ Dg for D ∈ ΓB(α).

We have λ(Dx) = (λ(D))g−1xg for D ∈ ΓB(α) and x ∈ G(B), and hence (ii) follows.

By Lemma 2.1.1(a), we have Gβ = g−1Gαg, and x 7→ g−1xg, for x ∈ Gα, defines an

isomorphism from Gα to Gβ . Since λ(Dx) = (Dx)g = (Dg)g−1xg = (λ(D))g

−1xg for

D ∈ ΓB(α) and x ∈ Gα, the assertion in (iii) then follows.

(c) Since B,C ∈ ΓB(α), α is adjacent to a vertex γ in B and a vertex δ in

C. So there exists g ∈ G such that (α, γ)g = (α, δ). This implies that g ∈ Gα

and Bg = C, and hence (Γ(α) ∩ B)g = Γ(α) ∩ C. Thus g induces a bijection

ρ from Γ(α) ∩ B to Γ(α) ∩ C defined by ρ : γ 7→ γg for γ ∈ Γ(α) ∩ B. By

Lemma 2.1.1(a), we have Gα,C = (Gα)C = (Gα)Bg = g−1(Gα)Bg = g−1(Gα,B)g, and

hence x 7→ g−1xg, for x ∈ Gα,B, defines an isomorphism from Gα,B to Gα,C . Since

ρ(γx) = (γx)g = (γg)g−1xg = (ρ(γ))g

−1xg, the action of Gα,B on Γ(α) ∩ B and the

action of Gα,C on Γ(α) ∩ C are permutationally isomorphic with respect to ρ. For

each x ∈ G[α], we have g−1xg ∈ G[α], and x 7→ g−1xg defines an automorphism of


G[α]. By a similar argument as above one can see that the actions of G[α] on Γ(α)∩B

and Γ(α) ∩ C are permutationally isomorphic with respect to ρ. 2

3.3 Refining the given partition

The main result of this section is the following theorem, which shows that in some

cases we can get a (nontrivial) refinement B∗ of the given G-invariant partition

B. For B∗, the parameters v∗, r∗, k∗, s∗ have analogous meanings to the parameters

v, r, k, s respectively for B.

Theorem 3.3.1 Suppose that Γ is a G-symmetric graph admitting a nontrivial G-

invariant partition B. Then Γ admits a second G-invariant partition B∗, which is

a refinement of B (possibly B∗ = B), such that the block size v∗ of B∗ is a common

divisor of v and k, and that s = cs∗, r∗ = cr for some integer c ≥ 1. The block of

B∗ containing α ∈ B is B∗ :=⋂

C∈ΓB(α)(Γ(C)∩B). In particular, if Gα is primitive

on ΓB(α), then either

(a) D(B) has no repeated blocks, or

(b) k divides v, B∗ = (Γ(C) ∩ B)g : g ∈ G (where C ∈ ΓB(B)), Γ is a

multicover of ΓB∗ with v∗ = k∗ = k, s∗ = s, r∗ = r, and Γ[B,C] ∼= Γ[B∗, C∗], where

B∗ ∈ B∗ and C∗ ∈ ΓB∗(B∗).

Proof Let B ∈ B. For a fixed vertex α ∈ B, we define a binary relation “ ∼α ” on

ΓB(α) by

E ∼α F ⇔ Γ(E) ∩ B = Γ(F ) ∩B

for E,F ∈ ΓB(α). Then clearly “ ∼α ” is an equivalence relation on ΓB(α). Let n

be the number of equivalence classes of “ ∼α ”, and let R(α) := Ri(α) : 1 ≤ i ≤ n

denote the partition of ΓB(α) induced by “ ∼α ”. Then, for g ∈ Gα and E,F ∈

ΓB(α), we have: E ∼α F ⇔ Γ(E) ∩ B = Γ(F ) ∩ B ⇔ Γ(Eg) ∩ B = Γ(F g) ∩ B

⇔ Eg ∼α Fg. This implies that R(α) is a Gα-invariant partition of ΓB(α). The

block size of R(α) is equal to the multiplicity m of D(B). Hence mn = r, and

in particular n is a divisor of r. Also, since E ∈ ΓB(α) implies Eg ∈ ΓB(αg)

and Γ(Eg) ∩ B = (Γ(E) ∩ B)g for g ∈ GB, from the definition of “ ∼α ” we have

R(αg) = (R(α))g = (Ri(α))g : 1 ≤ i ≤ n. Hence the size n of R(α) is independent

of the choice of α ∈ B.

Refining the Given Partition 35

Now we define an equivalence relation “ ∼ ” on B by

α ∼ β ⇔ ΓB(α) = ΓB(β) and R(α) = R(β)

for α, β ∈ B. Since, for g ∈ GB, α ∼ β implies ΓB(αg) = ΓB(βg) and R(αg) = R(βg),

the partition of B induced by “ ∼ ” is GB-invariant. Let α be a fixed vertex in B,

and let B∗ be the block of this partition containing α. Then Gα ≤ GB∗ ≤ GB, and

so (see the third paragraph of Section 2.2) B∗ := B∗g : g ∈ G is a G-invariant

partition of V (Γ) refining B. Thus v∗ = |B∗| = v/|GB : GB∗ | divides v. We

claim that B∗ =⋂

C∈ΓB(α)(Γ(C) ∩ B). In fact, let Cα1, . . . , Cαn be representatives

from R1(α), . . . , Rn(α), respectively. If β ∈⋂ni=1(Γ(Cαi) ∩ B), then for each i,

β ∈ Γ(Cαi) ∩ B = Γ(C) ∩ B for each C ∈ Ri(α), and hence ΓB(β) = ΓB(α) and

Cα1, . . . , Cαn are representatives from R1(β), . . . , Rn(β). This implies R(α) = R(β)

and hence β ∈ B∗. Conversely, for any β ∈ B∗, we have R(α) = R(β) and hence

R1(β), . . . , Rn(β) = R1(α), . . . , Rn(α). So β ∈⋂ni=1(Γ(Cαi) ∩ B). Thus we

have proved that B∗ =⋂ni=1(Γ(Cαi) ∩ B), that is, B∗ =

⋂

C∈ΓB(α)(Γ(C) ∩ B). Let

γ ∈ Γ(D) ∩ B, where D ∈ ΓB(α). Then there exists g ∈ GB such that γ = αg. So

γ ∈ B∗g =⋂

C∈ΓB(α)(Γ(C) ∩ B)g =⋂

C∈ΓB(γ)(Γ(C) ∩ B). Note that γ ∈ Γ(D) ∩ B

implies D ∈ ΓB(γ). So we have B∗g ⊆ Γ(D) ∩ B. Therefore, Γ(D) ∩ B is a union

of blocks of B∗. Hence v∗ is a divisor of k. Let c denote the number of blocks B∗g

of B∗ contained in D such that Γ(α) ∩ B∗g 6= ∅. Since GB is transitive on the flags

of D(B) (Lemma 3.2.5(b)), c is independent of the choice of D. Clearly, we have

s = cs∗. Since val(Γ) = rs = r∗s∗, this implies r∗ = cr.

Now we suppose thatGα is primitive on ΓB(α). If D(B) contains repeated blocks,

say C,D, let α ∈ Γ(C) ∩ B = Γ(D) ∩ B. Then each part Ri(α) of the Gα-invariant

partition R(α) of ΓB(α) has size at least 2. So the primitivity of Gα on ΓB(α) implies

that R(α) must be the trivial partition ΓB(α). Hence Γ(C)∩B is the same for all

C ∈ ΓB(α). Thus the equivalence relation “ ∼ ” on B defined above becomes: α ∼ β

if and only if Γ(E)∩B = Γ(F )∩B for any E ∈ ΓB(α), F ∈ ΓB(β). This implies that

the block of B∗ containing α is Γ(C) ∩ B, and hence B∗ = (Γ(C) ∩ B)g : g ∈ G.

Thus k = v∗ and so k is a divisor of v, and Γ is a multicover of ΓB∗ (that is, k∗ = v∗).

In this case, we have s∗ = s, r∗ = r and Γ[B,C] ∼= Γ[B∗, C∗] for adjacent blocks

B∗ := Γ(C) ∩ B, C∗ := Γ(B) ∩ C of B∗. 2

Theorem 3.3.1 implies the following known result.


Corollary 3.3.1 ([43, Lemma 3.3(c)]) Suppose that Γ is a G-locally primitive graph.

Suppose further that V (Γ) admits a nontrivial G-invariant partition B such that the

block size k of D(B) satisfies 2 ≤ k ≤ v − 1. Then either

(a) D(B) contains no repeated blocks, or

(b) k divides v, and V (Γ) admits a second nontrivial G-invariant partition B∗,

which is a refinement of B, such that Γ is a cover of ΓB∗.

Proof Since Γ is G-locally primitive, Corollary 3.2.1 applies. If the second possi-

bility in Corollary 3.2.1 occurs, then (a) above holds; otherwise we have s = 1 and

hence Gα is primitive on ΓB(α) by the G-local primitivity of Γ and Lemma 3.2.4.

In this latter case, from the second half of Theorem 3.3.1, either (a) or (b) above

holds. 2

Remark 3.3.1 We should emphasize that B∗ could be a trivial partition of V (Γ)

or identical with B in some cases. For example, if k = 1, then v∗ = 1 and of course

B∗ is trivial. However, there are other cases for which B∗ is a genuine refinement of

B, and these are the cases we are interested in. Corollary 3.3.1 above shows that

this is the case whenever Γ is G-locally primitive such that 2 ≤ k ≤ v− 1 and D(B)

contains repeated blocks.

A nontrivial G-invariant partition B of V (Γ) is said to be minimal if there is

no genuine refinement of B which is also a G-invariant partition of V (Γ). Any

imprimitive G-symmetric graph Γ admits at least one minimal nontrivial G-invariant

partition. Applying Theorem 3.3.1, the minimality of such a partition B implies

that either v∗ = 1 or v∗ = v. In the first case, we have⋂

C∈ΓB(α)(Γ(C) ∩ B) 6=⋂

C∈ΓB(β)(Γ(C) ∩ B) for distinct α, β ∈ B, and in this case we say that Γ is vertex-

distinct with respect to B. In the second case, since we have proved that v∗ is a

divisor of k, we must have v∗ = v = k and thus Γ is a multicover of ΓB. So Theorem

3.3.1 has the following consequence.

Corollary 3.3.2 Suppose that Γ is an imprimitive G-symmetric graph, and let B be

a minimal nontrivial G-invariant partition of V (Γ). Then either Γ is vertex-distinct

with respect to B, or Γ is a multicover of ΓB.

Chapter 4

The case k = v − 1: A generalanalysis

There is never a case when the root is in disorder and yet the

branches are in order.


In this chapter and the four chapters hereafter we will concentrate on the case

where k = v−1. This requirement is equivalent to the following: For distinct blocks

B,C ∈ B, either there are no edges between B and C, or there is a unique vertex

α ∈ B such that Γ(α) ∩ C = ∅. Thus in this case the design D(B) is degenerate,

with each (v − 1)-element subset of B occurring as a (possibly repeated) block of

D(B).

This chapter is devoted to a general analysis for the case k = v− 1. The results

obtained here will be used in the next four chapters and Chapter 11.

4.1 Notation and preliminary results

Let us first introduce some special notation for the case where k = v − 1. Suppose

Γ is a G-symmetric graph admitting a nontrivial G-invariant partition B such that

k = v − 1. Let B = B(α) for α ∈ V (Γ), and let

B(α) := C ∈ B : Γ(C) ∩B = B \ α. (4.1)

Thus B(α) is the set of blocks which are adjacent to B(α) in ΓB but contain no vertex

adjacent to α in Γ. It is easy to check that (B(α))g = Cg : C ∈ B(α) = B(αg) for

38 The case k = v − 1

any α ∈ V (Γ) and g ∈ G. If B(α) ∈ B(β) and B(β) ∈ B(α) hold simultaneously,

then we say that α, β are mates and that α is the mate of β in B(α). Clearly, if β

is the mate of α in B(β), then α is the mate of β in B(α). Let

A(α) := (B,C) : C ∈ B(α), (4.2)

the set of arcs of ΓB from B to an element of ΓB(B) containing no vertices adjacent

to α. Then we can view A(α) as a label attached to the vertex α. Denote

A(B) := A(α) : α ∈ B (4.3)

for a block B ∈ B, and set

A := A(α) : α ∈ V (Γ). (4.4)

Lemma 4.1.1 Suppose Γ is a G-symmetric graph which admits a G-invariant par-

tition B of V (Γ) such that k = v − 1 ≥ 1. Then A is a G-invariant partition

of Arc(ΓB), and hence G induces an action on A. The action of G on V (Γ) and

this action of G on A are permutationally equivalent with respect to the bijection

λ : α 7→ A(α). In particular, we have (A(α))g = A(αg) for α ∈ V (Γ), g ∈ G.

Proof Let α, β be distinct vertices of Γ. If B(α) 6= B(β), then the arcs in A(α)

and A(β) have different initial vertices; if B(α) = B(β), then B(α) ∩ B(β) = ∅ as

k = v − 1. In both cases, we get A(α) ∩ A(β) = ∅ and hence A is a partition of

Arc(ΓB). It is straightforward to show that this partition is a G-invariant partition

of Arc(ΓB), and hence G induces an action on A. Furthermore, the argument above

shows that λ : α 7→ A(α) is a bijection from V (Γ) to A. For any α ∈ V (Γ) and

g ∈ G, since B(αg) = (B(α))g, we have λ(αg) = A(αg) = (A(α))g = (λ(α))g.

Therefore, the actions of G on V (Γ) and A are permutationally equivalent with

respect to λ. 2

Next we define a graph Γ′ associated with (Γ,B) in the case where k = v − 1.

Definition 4.1.1 Let Γ′ be the graph with vertex set V (Γ) in which two vertices

α, β are adjacent if and only if they are mates (see Figure 1). In other words, α, β

are adjacent in Γ′ if and only if B(α), B(β) are adjacent in ΓB, α is the only vertex

in B(α) not adjacent to any vertex in B(β), and β is the only vertex in B(β) not

adjacent to any vertex in B(α).

Notations and Preliminaries 39

Note that (α, β) 7→ (B(α), B(β)) establishes a bijection from the set of arcs of

Γ′ to the set of arcs of ΓB.

rrα

rrβ

Γ

rrα

rrβ

Γ′

Figure 1 The definition of Γ′

Theorem 4.1.1 Suppose Γ is a G-symmetric graph which admits a nontrivial G-

invariant partition B such that k = v − 1 ≥ 1. Then the graph Γ′ defined above is

G-symmetric.

Proof Let (α, β), (γ, δ) be distinct arcs of Γ′. Then (B(α), B(β)), (B(γ), B(δ))

are distinct arcs of ΓB. Since ΓB is G-symmetric, there exists g ∈ G such that

(B(α), B(β))g = (B(γ), B(δ)), that is, (B(αg), B(βg)) = (B(γ), B(δ)). Since α is

the only vertex in B(α) not adjacent to any vertex in B(β), we know that αg is

the only vertex in B(αg) = B(γ) not adjacent to any vertex in B(βg) = B(δ), and

γ is the only vertex in B(γ) not adjacent to any vertex in B(δ). So we must have

αg = γ. Similarly, βg = δ. Hence (α, β)g = (γ, δ) and Γ′ is a G-symmetric graph.

2

We say that the graph Γ is vertex-distinguishable with respect to B if, for any

two adjacent blocks B,C of B and distinct vertices α, β ∈ Γ(B) ∩ C, we have

Γ(α) ∩ B 6= Γ(β) ∩ B. We conclude this section by proving the following lemma

which exemplifies graphs of this kind. This lemma will be used in Theorem 4.3.1(d)

and in the proof of Corollary 4.3.1.

Lemma 4.1.2 Suppose Γ is a G-symmetric graph admitting a nontrivial G-invariant

partition B. Then Γ is vertex-distinguishable with respect to B if, for adjacent blocks

B,C of B, one of the following conditions holds:

(a) Γ[B,C] is a matching;

(b) Γ[B,C] is a complete bipartite graph minus a perfect matching between the

vertices of Γ(C) ∩ B and Γ(B) ∩ C;

(c) GB,C acts primitively on Γ(B) ∩ C and Γ[B,C] 6∼= Kk,k.


Proof Clearly, the result is true whenever (a) or (b) occurs. Suppose that the

condition (c) is satisfied. If there exist distinct α, β ∈ Γ(B)∩C such that Γ(α)∩B =

Γ(β) ∩B, then γ ∈ C : Γ(γ) ∩B = Γ(α) ∩B is a block of imprimitivity for GB,C

in Γ(B) ∩ C and has size at least 2. Since this action is primitive, it follows that

Γ(γ) ∩ B = Γ(α) ∩ B for all γ ∈ Γ(B) ∩ C. This implies that Γ[B,C] ∼= Kk,k, a

contradiction. Thus, Γ is vertex-distinguishable with respect to B. 2

4.2 The case where k = 1 and v = 2

We will distinguish the following two cases:

I. k = v − 1 = 1; and

II. k = v − 1 ≥ 2.

In this section we discuss Case I, which can occur in a nontrivial way (see the

examples in [43, Section 5] and see also Theorem 5.1.3 and the remarks following

it). The characterization of Γ in Case I varies in difficulty according to the nature

of ΓB. For example, if ΓB = Cn, then r = 1 and Γ is uniquely determined (see

[43, Theorem 4.1(a)]), namely Γ = n ·K2, while if ΓB is a complete graph, then it

seems rather difficult to determine or describe Γ (see [43, Section 4]). In Section 8.3

we will give a general construction of imprimitive G-symmetric graphs with k = 1

and v ≥ 2. Here we prove some properties which hold only for the case where

k = v − 1 = 1.

Suppose then that k = v − 1 = 1. For each vertex α, let B(α) = α, α′ denote

the block of B containing α, so B(α) = B(α′). The adjacency relation for the graph

Γ′ defined in Definition 4.1.1 becomes: α and β are adjacent in Γ′ if and only if α′

and β ′ are adjacent in Γ. Besides Γ′, we can associate with Γ two other graphs Γ⋆

and Γ# (see Figure 2) defined as follows.

Definition 4.2.1 (a) Let Γ⋆ be the graph with vertex set V (Γ) in which α, β is

an edge if and only if either α, β or α′, β ′ is an edge of Γ;

(b) Let Γ# be the graph with vertex set V (Γ) such that α, β ′ and α′, β are

edges of Γ# if and only if either α, β or α′, β ′ is an edge of Γ.

The Case k = 1 and v = 2 41

rrα

α′

rrβ

β′

Γ

rrα

α′

rrβ

β′

Γ′

rrα

α′

rrβ

β′

Γ⋆

rrα

α′

rrβ

β′

Γ#

HH

HH

HH

Figure 2 The definitions of Γ′,Γ⋆ and Γ#

The graph Γ⋆ was defined in [43, Section 5] for a G-locally primitive graph

Γ. The following result is analogous to [43, Lemma 5.1] without assuming G-local

primitivity. It shows that the quotient graph ΓB may be covered by two (possibly

non-isomorphic) symmetric graphs. Let z be the involution which interchanges the

two vertices in each block of B.

Theorem 4.2.1 Suppose that Γ is a G-symmetric graph and B is a nontrivial G-

invariant partition of V (Γ) with block size v = k + 1 = 2. Then

(a) Γ′ ∼= Γ, and Γ′ is G-symmetric; and

(b) both Γ⋆ and Γ# are (G × 〈z〉)-symmetric, and B is a (G × 〈z〉)-invariant

partition of V (Γ). Also, Γ⋆B = Γ#B = ΓB and both Γ⋆ and Γ# are covers of ΓB.

Furthermore, if G is faithful on V (Γ), then it is faithful on B as well.

Proof By Theorem 4.1.1, Γ′ is G-symmetric, and the mapping z : α 7→ α′, for

α ∈ V (Γ), is an isomorphism from Γ to Γ′. Thus (a) is proved.

Clearly, 〈G, z〉 ∼= G×Z2. Since the edge set of Γ⋆ is the union of the sets of edges

of Γ and Γ′ it follows from (a) that G×〈z〉 ≤ Aut(Γ⋆) and that G×〈z〉 is transitive

on the arcs of Γ⋆. Also, B is a (G × 〈z〉)-invariant partition of V (Γ), and Γ⋆ is a

cover of Γ⋆B = ΓB. Moreover, ΓB = Γ#B , and Γ# is a cover of ΓB. For two adjacent

blocks B = α, α′ and C = β, β ′ of ΓB, suppose that (α, β) is an arc of Γ. Then

(α, β ′) and (β, α′) are arcs of Γ# which are interchanged by z. It is also easy to

check that G preserves the edge set of Γ#. It follows that G × 〈z〉 is transitive on

the arcs of Γ#.

Let B(α) = α, α′ be a block of B. If g ∈ G is any element which maps α to α′,

then g interchanges α and α′. Hence g interchanges ΓB(α) and ΓB(α′). Note that

ΓB(α) and ΓB(α′) are disjoint since k = 1. Thus, g acts nontrivially on B. It follows

that, if G is faithful on V (Γ), then G is also faithful on B. 2


Remark 4.2.1 The graphs Γ⋆, Γ# defined in Definition 4.2.1 may, or may not, be

isomorphic to each other. For example, under the conditions of Theorem 4.2.1, if

ΓB = C4, then both Γ⋆ and Γ# are 2 · C4; while if ΓB = C3, then Γ⋆ = C6 whilst

Γ# = 2 · C3. So Γ⋆ and Γ# may be non-isomorphic covers of ΓB.

4.3 A general discussion: k = v − 1 ≥ 2

In the remaining sections of this chapter we investigate the general case where

v = k + 1 ≥ 3. Note that if, in addition, Γ is G-locally primitive, then D(B)

contains no repeated blocks (by Corollary 3.3.1, noting that k does not divide v

here). This however is not true in general, that is, the multiplicity of D(B) can be

greater than one for general symmetric graphs with v = k + 1 ≥ 3.

Theorem 4.3.1 Suppose that Γ is a G-symmetric graph and B is a nontrivial G-

invariant partition of V (Γ) with block size v = k+ 1 ≥ 3. Let B be a block of B and

α ∈ B. Then the following (a)-(d) hold.

(a) D(B) has v distinct blocks and the multiplicity m of D(B) is equal to |B(α)|,

so b = mv, r = m(v − 1), and D(B) is a 2-(v, v − 1, m(v − 2))-design.

(b) Gα has two orbits on ΓB(B), namely, B(α) and ΓB(B) \ B(α).

(c) G[B] ≤ G(B) and equality holds whenever D(B) contains no repeated blocks.

Moreover, if G is faithful on V (Γ), then it is also faithful on B.

(d) If G is faithful on V (Γ), D(B) contains no repeated blocks, ΓB is connected

and Γ is vertex-distinguishable with respect to B, then GB acts faithfully on B and

ΓB(B).

Proof (a) Since GB is transitive on B, each (v − 1)-subset of B is the trace of a

block of D(B) and hence D(B) has v distinct blocks each repeated m times. So

we have m = |B(α)| and b = mv. This, together with vr = bk = b(v − 1), gives

r = m(v − 1). In particular, D(B) is a 2-(v, v − 1, m(v − 2))-design.

(b) Clearly, B(α) is Gα-invariant. Let C,D ∈ B(α). Since ΓB is G-symmetric,

there exists g ∈ G with Bg = B,Cg = D. Now αg = α for otherwise α is adjacent

to no vertex in C but αg is adjacent to at least one vertex in Cg = D. Thus,

g ∈ Gα and hence Gα is transitive on B(α). Now let C,D ∈ ΓB(B) \ B(α). Then

α ∈ Γ(C) ∩ Γ(D) ∩ B. So there exist β ∈ C, γ ∈ D which are adjacent to α. Since

The Case k = v − 1 ≥ 2 43

Γ is G-symmetric, there exists g ∈ G with (α, β)g = (α, γ). Thus, g ∈ Gα and

Cg = D. So ΓB(B) \ B(α) is a Gα-orbit.

(c) If g ∈ G[B] then, for each β ∈ B, g fixes setwise each block C ∈ B(β) and

hence fixes setwise Γ(C) ∩ B. Therefore, g fixes B \ (Γ(C) ∩ B) = β. Thus, we

have G[B] ≤ G(B). Moreover, if g ∈ G fixes setwise each block of B, then it lies in

G[B] for each B, and hence fixes each vertex of Γ. This implies g = 1 provided that

G is faithful on V (Γ). So, if G is faithful on V (Γ), then it is faithful on B. Suppose

that D(B) contains no repeated blocks and g ∈ G(B). Then for each α ∈ B, g fixes

the unique block in B(α), and hence g fixes each block of ΓB(B) setwise. So g ∈ G[B]

and thus G[B] = G(B).

(d) From (c) and the assumption that D(B) contains no repeated blocks, we

have G(B) = G[B]. Let g ∈ G(B) = G[B]. Then for C ∈ ΓB(B), g fixes the unique

vertex in C \ (Γ(B) ∩ C), and for each β ∈ Γ(B) ∩ C, we have βg ∈ Γ(B) ∩ C and

Γ(β) ∩B = Γ(βg) ∩B (since g fixes B pointwise). Since Γ is vertex-distinguishable

with respect to B, we get βg = β. Thus g ∈ G(C) and hence G(B) ≤ G(C). By a

similar argument G(C) ≤ G(B), so G(B) = G(C). Since ΓB is connected, this equality

is true for any two blocks B,C (not necessarily adjacent), and hence G(B) = 1 = G[B]

since G is assumed to be faithful on V (Γ). Thus, GB is faithful on B and on ΓB(B).

2

By Lemma 4.1.1 the induced action of G on A can be defined by (A(α))g =

A(αg), for α ∈ V (Γ), g ∈ G, and this action is permutationally equivalent to the

action of G on V (Γ). Clearly, A(B) is a GB-invariant subset of A. Thus, GB

induces an action on A(B). Also, one can see that B(B) := B(α) : α ∈ B is a

GB-invariant partition of ΓB(B), and hence GB induces an action on B(B) defined

by (B(α))g = B(αg) for α ∈ B and g ∈ GB. The following theorem will play an

important role in our later discussion. It shows in particular that the actions of GB

on B, A(B) and B(B) are permutationally equivalent and doubly transitive.


invariant partition B with block size v = k+1 ≥ 3. Let B ∈ B, α ∈ B and C ∈ B(α).

Then the following (a)-(c) hold.

(a) The action of GB on B is permutationally equivalent to the actions of GB

on A(B) and B(B) with respect to the bijections defined by α 7→ A(α), α 7→ B(α),


for α ∈ B, respectively.

(b) GB,C is transitive on B \ α, A(B) \ A(α) and B(B) \ B(α). In par-

ticular, the actions of GB on B, A(B) and B(B) are doubly transitive.

(c) GB,C = Gα,C = Gδ,B = Gα,δ, where δ is the mate of α in C.

Proof (a) Clearly, the actions of GB on B(B) and A(B) are permutationally

equivalent with respect to the natural bijection B(α) 7→ A(α), for α ∈ B. The

permutation equivalence of the actions of GB on B and A(B) follows immediately

from Lemma 4.1.1.

(b) First, since GB is transitive on B, from (a) above GB is transitive on A(B)

and B(B). Second, since v = k + 1 ≥ 3, for distinct vertices β, γ ∈ B \ α

there exist ε, η ∈ C \ δ which are adjacent in Γ to β, γ respectively, where δ

is the mate of α in C. By the G-symmetry of Γ, there exists g ∈ G such that

(β, ε)g = (γ, η). This implies g ∈ GB,C and (A(β))g = A(γ). HenceGB,C is transitive

on A(B)\A(α). Since by (a) above the actions of GB,C on B\α, A(B)\A(α)

and on B(B) \ B(α) are permutationally equivalent, it follows that GB,C is also

transitive on B \ α and B(B) \ B(α). Note that, since (B,C) ∈ A(α) and A is

a G-invariant partition of Arc(ΓB) (Lemma 4.1.1), GB,C is a subgroup of the setwise

stabilizer (GB)A(α) of A(α) in GB. So (GB)A(α) is transitive on A(B) \ A(α).

Therefore, we conclude that GB is doubly transitive on A(B) and hence doubly

transitive on B and B(B).

(c) Clearly, we have Gα,C ≤ GB,C since an element of G fixing α must fix B

setwise. Conversely, if g ∈ G fixes B and C setwise, then it must fix the unique

vertex α of B not adjacent to any vertex of C. So we have GB,C ≤ Gα,C and hence

GB,C = Gα,C . Similarly, one can show GB,C = Gδ,B and GB,C = Gα,δ. 2

Now let us consider the case where, in addition to our assumption k = v−1 ≥ 2,

Γ is G-locally primitive and ΓB is connected. In such a case, since k does not divide

v, from Corollary 3.3.1 we know that (i) D(B) contains no repeated blocks, and

hence b = v ≥ 3 and r = v − 1 by Theorem 4.3.1(a). Also, Corollary 3.2.1 implies

that (ii) Γ[B,C] ∼= (v − 1) ·K2 is a matching. From (ii) and Lemma 4.1.2 we know

that Γ is vertex-distinguishable with respect to B, and hence GB is faithful on B

provided that G is faithful on V (Γ) (Theorem 4.3.1(d)). Also from (i) and (ii) we

know that, for each α ∈ B, there exists a bijection from B \ α to Γ(α), namely

The Case k = v − 1 ≥ 2 45

each β ∈ B \ α corresponds to the unique neighbour of α in the unique block

of B(β). So Gα is primitive on B \ α as Gα is primitive on Γ(α). Therefore,

GB is doubly primitive on B, and hence is doubly primitive on ΓB(B) by Theorem

4.3.2(a). So we deduce the following result which was proved in [43, Theorem 5.3].

Corollary 4.3.1 Suppose that Γ is a G-locally primitive graph admitting a nontriv-

ial G-invariant partition B such that v = k+1 ≥ 3 and ΓB is connected. Then b = v,

r = v − 1, and the actions of GB on B and ΓB(B) are permutationally equivalent,

and doubly primitive. Moreover, these actions are faithful if in addition G is faithful

on V (Γ).

Now we consider the graph Γ′ defined in Definition 4.1.1. Each maximal clique

of Γ′ has at most m+1 vertices since the valency of Γ′ is m, where m = |B(α)|. The

following result shows that if each maximal clique of Γ′ does contain m+1 vertices,

or equivalently if Γ′ ∼= ℓ · Km+1 for some ℓ, then we obtain a second G-invariant

partition of V (Γ). This condition holds in particular when m = 1, and the following

result will be used in this case in the next chapter.


invariant partition B with blocks of size v = k + 1 ≥ 3. Let α ∈ V (Γ). Then

P = (α∪Γ′(α))g : g ∈ G is a G-invariant partition of V (Γ) if and only if V (Γ)

is a disjoint union of (m+ 1)-cliques of Γ′, where m = |B(α)|.

Proof Set Γ′(α) = α1, α2, . . . , αm and B′ = α ∪ Γ′(α), and suppose that V (Γ)

is a disjoint union of (m+ 1)-cliques of Γ′. Then B′ is the unique (m+ 1)-clique of

Γ′ containing α. Since G permutes the connected components of Γ′, it follows that

P is a G-invariant partition of V (Γ).

Conversely, suppose P is a G-invariant partition of V (Γ). For any i, 1 ≤ i ≤ m,

let g ∈ G be such that αg = αi. Then B′g = B′ since αi is in both B′ and B′g,

and hence Γ′(αi) = Γ′(αg) = (Γ′(α))g = (B′ \ α)g = B′ \ αi. Therefore, B′ is

a clique of Γ′ with the maximum possible size m + 1. In other words, V (Γ) is a

disjoint union of (m+ 1)-cliques of Γ′. 2


4.4 Analysing an extreme case

Let n := |B(α) ∩ B(β)| for adjacent vertices α, β of Γ. Since Γ is G-symmetric,

this parameter n is independent of the choice of such α and β. Clearly, we have

0 ≤ n ≤ m with the extreme case n = 0 occurring only if girth(ΓB) ≥ 4, where

m = |B(α)| is the multiplicity of D(B). We will study this extreme case in the next

chapter under the additional assumption m = 1 (see Theorems 5.1.2 and 5.1.3). In

this section we study the second extreme case where n = m, that is, the case where

B(α) = B(β) for adjacent vertices α, β of Γ. Our study below shows that in this

case all of Γ, ΓB and Γ[B,C] can be determined explicitly. We first give examples

of symmetric graphs with this property.

Example 4.4.1 (a) Let X be a 3-transitive group acting on a finite set I of degree

v + 1 ≥ 4. Let Γ be the graph with vertex set V := I(2) in which (i, h), (i′, h′) are

adjacent if and only if i 6= i′ and h = h′. Then the 3-transitivity of X implies that

Γ is X-symmetric and admits the X-invariant partition B := i : i ∈ I, where i

consists of members of V with first coordinate i. Clearly, we have k = v − 1 ≥ 2,

Γ ∼= (v+1) ·Kv, ΓB∼= Kv+1, Γ[B,C] ∼= (v−1) ·K2 for adjacent blocks B,C of B, and

D(B) contains no repeated blocks. Also, for adjacent vertices α = (i, h), α′ = (i′, h)

of Γ, we have B(α) = B(α′) = h.

(b) Now let us consider the case where the multiplicity m ≥ 2. Let X and I

be as in (a) above and let Y be a 2-transitive group acting on a finite set J of

degree m. Then G := X × Y is transitive on V := I(2) × J in its action defined by

(i, h, j)(x,y) := (ix, hx, jy) for (i, h, j) ∈ V and (x, y) ∈ G. Define the graph Γ with

vertex set V in which (i, h, j), (i′, h′, j′) are adjacent if and only if i 6= i′ and h = h′.

Then Γ ∼= (v + 1) ·Kvm, and the assumptions on X, Y imply that Γ is G-symmetric.

Clearly, Γ admits B := [i, j] : i ∈ I, j ∈ J as a G-invariant partition, where

[i, j] := (i, h, j) : h ∈ I \ i. We have ΓB∼= Kv+1

m with [i, j], [i′, j′] adjacent if and

only if i 6= i′. Also, we have Γ[B,C] ∼= (v−1)·K2 for adjacent blocks B,C of B (hence

k = v − 1 ≥ 2), and the multiplicity of D(B) is equal to m. Moreover, for adjacent

vertices α = (i, h, j), α′ = (i′, h, j′) of Γ, we have B(α) = B(α′) = [h, ℓ] : ℓ ∈ J.

In the following theorem, we will show that the graphs Γ in Example 4.4.1 are

the only G-symmetric graphs with ΓB connected such that k = v − 1 ≥ 2 and

An Extreme Case 47

B(α) = B(β) for adjacent vertices α, β of Γ, and that ΓB, Γ[B,C] are as shown in

this example. First, we have the following simple observation, which shows that the

fact Γ[B,C] ∼= (v − 1) ·K2 in Example 4.4.1 is not a coincidence.

Lemma 4.4.1 Suppose that Γ is a G-symmetric graph admitting a nontrvial G-

invariant partition B such that k = v − 1 ≥ 2. Let m be the multiplicity of D(B),

and let n = |B(α) ∩ B(β)| for (α, β) ∈ Arc(Γ), as defined above. Then we have

sn ≤ m,

where, recall that, s = |Γ(α) ∩ C| (for a flag (α,C) of D(B)) is the valency of

Γ[B,C]. In particular, if n > m/2, then Γ[B,C] ∼= (v − 1) ·K2.

Proof Let B ∈ B and α ∈ B. Let C ∈ ΓB(α) and set Γ(α) ∩ C = β1, . . . , βs.

Then B(α) ∩B(βi), for i = 1, . . . , s, are pairwise disjoint and each of them contains

n blocks of B(α). So we have sn ≤ m. In particular, if n > m/2, then we must have

s = 1 and thus Γ[B,C] ∼= (v − 1) ·K2. 2


invariant partition B such that k = v− 1 ≥ 2. Suppose further that ΓB is connected

and that B(α) = B(β) for adjacent vertices α, β of Γ. Let m be the multiplicity of

D(B). Then Γ ∼= (v + 1) · Kvm, ΓB

∼= Kv+1m , Γ[B,C] ∼= (v − 1) · K2 for adjacent

blocks B,C, and the induced action of G on the natural (v + 1)-partition B of ΓB

is 3-transitive (thus (ΓB)B ∼= Kv+1 is (G, 2)-arc transitive). Moreover, the vertices

of Γ can be labelled by ordered triples of integers such that the following (a)-(c) hold

(where we set I := 0, 1, . . . , v and J := 1, 2, . . . , m):

(a) V (Γ) = I(2) × J , and two vertices (i, h, j), (i′, h′, j′) ∈ V (Γ) are adjacent in

Γ if and only if i 6= i′ and h = h′.

(b) B = [i, j] : i ∈ I, j ∈ J, where [i, j] := (i, h, j) : h ∈ I \ i, and

[i, j], [i′, j′] are adjacent blocks if and only if i 6= i′.

(c) B = i : i ∈ I, where i = [i, j] : j ∈ J.

Conversely, the graph Γ defined in (a) together with the group G = X×Y satisfies

all conditions of the theorem, where X is a group acting 3-transitively on I, Y is a

group acting 2-transitively on J whenever m ≥ 2, and the action of G on V (Γ) is

as defined in Example 4.4.1.


Proof By our assumption we have n = m > m/2. Thus Lemma 4.4.1 implies

(1) Γ[D,E] ∼= (v − 1) ·K2 for adjacent blocks D,E of B.

Let B be a block of B and let α1, α2, . . . , αv be vertices of B. For each αi ∈ B, we

label (in an arbitrary way) the m blocks in B(αi) by [i, j], j ∈ J . Also, we label the

unique mate βij of αi in the block [i, j] by (i, 0, j), j ∈ J . For each block [i, j] and

for each h ∈ I \ 0 distinct from i, (1) implies that [i, j] contains a unique vertex

adjacent to αh. We label such a vertex in [i, j] by (i, h, j). In view of (1) one can

see that each vertex in [i, j] receives a unique label, and that the labels of distinct

vertices in [i, j] have distinct second coordinates. Therefore, for each i ∈ I \0 and

j ∈ J , we may identify the block [i, j] with the set (i, h, j) : h ∈ I \ i. By our

assumption, for i, h ∈ I \ 0 with i 6= h and j ∈ J , we have

(2) B((i, h, j)) = B(αh) = [h, 1], [h, 2], . . . , [h,m].

In particular, this implies that

(3) [i, j], [i′, j′] are adjacent blocks, for distinct i, i′ ∈ I \ 0 and any j, j′ ∈ J .

Moreover, if two vertices (i, h, j), (i′, h′, j′) are adjacent, where i, i′, h, h′ ∈ I \0

with i 6= h, i′ 6= h′ and j, j′ ∈ J , then by (2) and our assumption we must have

B(αh) = B((i, h, j)) = B((i′, h′, j′)) = B(αh′), which is true only when h = h′. This,

together with (1) and (3), implies the following assertion.

(4) For distinct i, i′ ∈ I \ 0 and any j, j′ ∈ J , two labelled vertices (i, h, j),

(i′, h′, j′) of Γ are adjacent if and only if h = h′. In other words, for adjacent

blocks D = [i, j], E = [i′, j′] of B, the bipartite subgraph Γ[D,E] of Γ is the

(v − 1)-matching with edges joining (i, h, j) and (i′, h, j′), for h ∈ I \ i, i′.

Therefore, (i, i′, j) and (i′, i, j′) are mates and hence, for the graph Γ′ defined in

Definition 4.1.1, we have

(5) Γ′((i, h, j)) = (h, i, j′) : j′ ∈ J.

Now let us examine a particular labelled vertex, say (i, h, j). From Theorem

4.3.1(a) and (1) above, the valency of Γ is m(v − 1), and hence the neighbourhood

Γ((i, h, j)) of (i, h, j) contains m(v − 1) vertices. From (4) we have (i′, h, j′) :

An Extreme Case 49

i′ ∈ I \ 0, h, i, j′ ∈ J ⊆ Γ((i, h, j)) and this contributes m(v − 2) neighbours of

(i, h, j). Note that αh is also a neighbour of (i, h, j). Apart from these, there are

m− 1 remaining neighbours of (i, h, j), which we denote by δ2, . . . , δm, respectively.

By (1) these vertices δ2, . . . , δm belong to distinct blocks, say B2, . . . , Bm, of B. For

each δt, we have B(δt) = B((i, h, j)) = B(αh) = [h, 1], [h, 2], . . . , [h,m] by (2) and

our assumption. In particular, this implies that all the blocks [h, ℓ], for ℓ ∈ J ,

are adjacent to the block Bt. On the other hand, from (5) we have Γ′((h, h′, ℓ)) =

(h′, h, t) : t ∈ J for each vertex (h, h′, ℓ) ∈ [h, ℓ] \ βhℓ. In other words, the m

mates of each vertex in [h, ℓ]\βhℓ are in⋃

h′∈I\0,h,t∈J [h′, t]. So the only possibility

is that βhℓ is the mate of δt in [h, ℓ], for each ℓ ∈ J . Consequently, we have

(6) B(βh1) = · · · = B(βhm) = B,B2, . . . , Bm, and hence none of B,B2, . . . , Bm

coincides with [i, j] for any i ∈ I \ 0, j ∈ J .

We know from (3) that the blocks [i′, j′], for i′ ∈ I \ 0, h and j′ ∈ J , are all

adjacent to [h, ℓ]. Besides these m(v − 1) blocks, B,B2, . . . , Bm are the only blocks

of B adjacent to [h, ℓ] in ΓB since ΓB has valency mv (Theorem 4.3.1(a)). Therefore,

if we apply the procedure above to another vertex (i′, h, j′), we would get the same

blocks B2, . . . , Bm. In other words, these blocks are independent of the choice of

the vertex (i, h, j) (depending only on h), and hence they are adjacent to the block

[i, j] for any i ∈ I \ 0 and j ∈ J . Moreover, since the mate δt of βhℓ in Bt is

unique, the vertices δ2, . . . , δm are also independent of the choice of (i, h, j) and thus

they are common neighbours of all such vertices (i, h, j). Thus, since the valency

of ΓB is mv, B,B2, . . . , Bm are the only unlabelled blocks of B. From this and by

a similar argument to that above, we see that for each h ∈ I \ 0, all the vertices

(i, h, j), i ∈ I \ 0, h, j ∈ J , have a common neighbour in each Bt, which we now

label by (0, h, t). Since for distinct h, h′ the vertices (i, h, j), (i, h′, j) have different

neighbours in Bt, the vertices of Bt receive pairwise distinct labels. Now let us

label B,B2, . . . , Bm with [0, 1], [0, 2], . . . , [0, m], respectively, and label each αh with

(0, h, 1). Then all the vertices of Γ and all the blocks of B have been labelled. From

the labelling above, the validity of (a) and (b) follows immediately.

Since the valency of Γ is m(v− 1), the argument above also shows that for each

h ∈ I the connected component of Γ containing the vertex αh is the complete v-

partite graphKvm with v-partition (i, h, j) : j ∈ J : i ∈ I, where we set α0 = β11.


Hence we have Γ ∼= (v + 1) · Kvm. Also, ΓB is the complete (v + 1)-partite graph

Kv+1m with (v + 1)-partition B := i : i ∈ I, where i := B(αi) = [i, j] : j ∈ J

for i ∈ I. Clearly, (ΓB)B ∼= Kv+1 and B is a G-invariant partition of B. From

Theorem 4.3.2(b), GB is doubly transitive on B(γ) : γ ∈ B. The setwise stabilizer

in G of the block 0 contains GB as a subgroup, and so is doubly transitive on the

neighbourhood B \ 0 of 0 in (ΓB)B. Therefore, (ΓB)B is (G, 2)-arc transitive and

hence G is 3-transitive on B.

Finally, for G = X × Y with X triply transitive on I and Y doubly transitive

on J whenever m ≥ 2, Example 4.4.1 shows that the graph Γ defined in (a) satisfies

all the conditions in the theorem. 2

Remark 4.4.1 In Theorem 4.4.1, G may or may not be faithful on B. (This can be

seen from Example 4.4.1, where the action of G on B is permutationally equivalent

to the action of X on I which is not necessarily faithful.) Let K be the kernel of the

action of G on B, and set H := G/K. Then H is 3-transitive and faithful on B of

degree v+1, and G is an extension of K by H . From the classification of finite highly

transitive permutation groups (see Theorem 2.1.1 and the comments following it), H

is one of the following: Sv+1 (v ≥ 3), Av+1 (v ≥ 4), Mv+1 (v = 10, 11, 21, 22, 23), M11

(v = 11), AGL(d, 2) (v = 2d − 1), Z42.A7 (v = 15), and PSL(2, v) ≤ H ≤ PΓL(2, v)

(v a prime power). Example 4.4.1 shows that the multiplicity m of D(B) can be

any positive integer and H can be any group listed above.

Chapter 5

The case k = v − 1 ≥ 2: D(B)contains no repeated blocks

What is most perfect seems to be incomplete; but its utility is

unimpaired. What is most full seems to be empty; but its usefulness is

inexhaustible. What is most straight seems to be crooked. The greatest

skills seems to be clumsy. The greatest eloquence seems to stutter.

Lao Tzu (6th or 4th Cent. B.C. ?), Tao Te Ching 45

In this and the next two chapters we continue our study for the case where

k = v − 1 ≥ 2 under the additional assumption that D(B) contains no repeated

blocks (that is, the multiplicity of D(B) is equal to 1). Not only is this a natural

assumption geometrically, but also we will prove (Theorem 5.1.2) that it occurs if

and only if ΓB is (G, 2)-arc transitive. In this case each vertex of Γ can be labelled in

a natural way by an arc of ΓB. Inspired by this labelling we then give a very simple

and elegant method for constructing all such graphs. In particular, our construction

shows that such a graph Γ can be reconstructed from the quotient ΓB and the action

of G on B.

5.1 The case D(B) contains no repeated blocks

In the following we suppose that Γ is a G-symmetric graph admitting a nontrivial

G-invariant partition B such that k = v − 1 ≥ 2 and D(B) has no repeated blocks.

Then the valency of the graph Γ′ (defined in Definition 4.1.1) is 1 and thus each

52 No repeated blocks

vertex α ∈ V (Γ) has a unique mate α′, namely the unique vertex adjacent to α

in Γ′. Hence the blocks of the partition P defined in Theorem 4.3.3 are the pairs

α, α′, and the map z : α 7→ α′ defines a G-invariant bijection on V (Γ). So A(α)

contains only one arc (B(α), B(α′)) and, by Theorem 4.3.1(b), B(α′) is the unique

block in ΓB(B) fixed setwise by Gα. As in the G-locally primitive case [43], the

mapping λ : α 7→ A(α) of Lemma 4.1.1 defines, for each α ∈ V (Γ), a unique label

“B(α)B(α′)” for α with the blocks of B containing α and α′ as the first and the

second coordinates, respectively. Set B∗ = Bz = “CB” : C ∈ ΓB(B) for B ∈ B.

Then it follows from the definition that no vertex in B∗ is adjacent to any vertex in

B, that is, B∗ ∩ Γ(B) = ∅. Thus no neighbour of α ∈ B has a label involving B as

either coordinate.

Theorem 5.1.1 Suppose that Γ is a G-symmetric graph, B is a nontrivial G-

invariant partition of V (Γ) with block size v = k+1 ≥ 3 such that D(B) contains no

repeated blocks. Then ΓB has valency b = v. Let z : α 7→ α′, α ∈ V (Γ), as defined

above. Then

(a) the actions of G on V (Γ) and on the set of arcs of ΓB are permutationally

equivalent, and each α ∈ V (Γ) can be uniquely labelled by a pair “BB′” of adjacent

blocks of B, where B = B(α) and B′ is the unique block in ΓB(B) fixed setwise by

Gα;

(b) z centralises G and is an involution (that is, z2 = 1), and z 6∈ G; also

P = α, α′ : α ∈ V (Γ) is a (G× 〈z〉)-invariant partition of V (Γ);

(c) B∗ := (B∗)g : g ∈ G is a G-invariant partition of V (Γ) with blocks of size

v; and GB∗ = GB is doubly transitive on B and B∗.

Proof Theorem 4.3.1(a) implies that b = v. Each A(α) can be identified with the

arc (B(α), B(α′)) of ΓB and each arc of ΓB has this form. So it follows from Lemma

4.1.1 that the actions of G on V (Γ) and on the set of arcs of ΓB are permutationally

equivalent. Clearly, z is an involution and leaves P invariant. This and Theorem

4.3.3 together imply that P is a (G × 〈z〉)-invariant partition of V (Γ). For each

g ∈ G and “BD” ∈ V (Γ), we have “BD”zg = “DB”g = “DgBg” = “BgDg”z =

“BD”gz and hence z centralises G. If B∗ ∩ (B∗)g 6= ∅ for some g ∈ G then, since

(B∗)g = “CgBg” : “CB” ∈ B∗ = (Bg)∗, we have B∗ ∩ (Bg)∗ 6= ∅, which implies

g ∈ GB and consequently (B∗)g = B∗. Thus, B∗ is a G-invariant partition of V (Γ)

No Repeated Blocks 53

with block size v. Since z interchanges B and B∗ whilst G leaves B invariant, it

follows that z 6∈ G. Clearly, GB∗ = GB and the actions of GB on B and B∗ are

permutationally equivalent with respect to z : α 7→ α′. So by Theorem 4.3.2(b), GB

is doubly transitive on both B and B∗. 2

Corollary 5.1.1 Suppose that Γ is a G-symmetric graph with B a nontrivial G-

invariant partition of block size v = k + 1 ≥ 3 such that D(B) contains no repeated

blocks. Then for any 2-arc (B,C,D) of ΓB, we have

(a) G“CB” = G“BC” = GB,C and hence G“CB”,D = GB,C,D; and

(b) the actions of GB,C,D on D \ “DC” and ΓB(D) \ C are permutationally

equivalent.

Proof By Theorem 5.1.1(a) we have G“CB” = G“BC” = GB,C , and hence G“CB”,D =

GB,C,D. Note that GB,C,D (≤ GD) fixes “DC” and fixes C setwise. Also, Theorem

4.3.2 implies that the actions of GD on D and ΓB(D) are permutationally equivalent

with respect to the bijection ρ : “DE” 7→ E for E ∈ ΓB(D). Therefore, GB,C,D

induces actions on D\“DC” and ΓB(D)\C, respectively, and these two actions

are permutationally equivalent with respect to the restriction of ρ to D \ “DC”.

2

As advertised at the beginning of this chapter, we now prove that, if k = v−1 ≥ 2,

then D(B) contains no repeated blocks if and only if ΓB is (G, 2)-arc transitive.

Theorem 5.1.2 Suppose that Γ is a G-symmetric graph, and B is a nontrivial G-

invariant partition of V (Γ) with block size v = k + 1 ≥ 3. Then D(B) contains no

repeated blocks if and only if ΓB is (G, 2)-arc transitive. Furthermore, in this case

either

(a) adjacent vertices have labels involving four distinct blocks, or

(b) there exist two adjacent vertices of Γ which share the same second coordinate.

In this case, Γ[B,C] ∼= (v−1) ·K2, Γ[B∗] ∼= Kv, Γ ∼= n(v+1) ·Kv and ΓB∼= n ·Kv+1

for some integer n ≥ 1, and the group induced by G on the connected component

B ∪ ΓB(B) of ΓB is 3-transitive. In particular, if ΓB is connected, then Γ ∼=

(v + 1) · Kv, ΓB∼= Kv+1 and G acts on B as a 3-transitive permutation group of

degree v + 1.


Proof Suppose D(B) has no repeated blocks. Then for each α ∈ B, B(α) can

be identified with the unique block it contains, and thus B(B) can be identified

with ΓB(B). So Theorem 4.3.2(b) implies that GB is doubly transitive on ΓB(B).

Since G is transitive on B, it follows from Lemma 3.1.1(b) that ΓB is (G, 2)-arc

transitive. Conversely suppose that ΓB is (G, 2)-arc transitive, and let α, β, γ be

pairwise distinct vertices of B. (Note that v ≥ 3.) If D(B) contains repeated

blocks, then there are distinct blocks C1, C2 ∈ B(α). Let D ∈ B(β) and E ∈ B(γ).

By the (G, 2)-arc transitivity of ΓB there exists g ∈ GB with (C1, C2)g = (D,E).

Note that, as mentioned before Theorem 4.3.2, B(B) = B(δ) : δ ∈ B is a GB-

invariant partition of ΓB(B). So Cg1 = D implies (B(α))g = B(β), whilst Cg

2 = E

implies (B(α))g = B(γ). This contradiction shows that D(B) contains no repeated

blocks. Thus the first assertion is proved.

For the rest of the proof we assume that D(B) has no repeated blocks. If adjacent

vertices of Γ have different second coordinates, then it follows from the definition of

the labels that two adjacent vertices of Γ have labels involving four distinct blocks.

Suppose there exist two adjacent vertices whose second coordinates are the same.

Since G acts transitively on B, we may assume without loss of generality that there

are two adjacent vertices in B∗. Since GB∗ is doubly transitive on B∗ (Theorem

5.1.1(c)), it follows that B∗ induces a complete graph Kv. Since Γ is G-symmetric

and since B∗ is a G-invariant partition of V (Γ) (Theorem 5.1.1(c)), it follows that

each edge of Γ joins two vertices in the same block of B∗. This means that each

block of B∗ induces a connected component Kv of Γ and hence Γ = |B∗| ·Kv. This

implies in particular that Γ[B,C] is a matching of v − 1 edges. Note that any two

blocks in ΓB(B) are adjacent in ΓB and hence B ∪ ΓB(B) induces a complete

subgraph Kv+1 of ΓB. Since the valency of ΓB is b = v, the subgraph induced by

B ∪ ΓB(B) is a connected component of ΓB. This implies (i) ΓB = n ·Kv+1, and

hence Γ = n(v+1) ·Kv, where n is the number of connected components of ΓB; and

(ii) since G is transitive on B and GB is doubly transitive on ΓB(B), as shown above,

it follows that the group induced on the connected component B ∪ ΓB(B) of ΓB

is 3-transitive. In particular, if ΓB is connected, then ΓB = Kv+1, Γ = (v + 1) ·Kv

and G is 3-transitive on B = B ∪ ΓB(B) with degree |B| = v + 1. 2

No Repeated Blocks 55

Remark 5.1.1 Under the assumption that D(B) contains no repeated blocks, two

adjacent vertices α, β of Γ share the same second coordinate if and only if the size

of B(α) ∩ B(β) is equal to 1. Hence we can also prove the assertions in part (b) of

Theorem 5.1.2 by applying Theorem 4.4.1 to each connected component of ΓB. If G

is faithful on V (Γ) and ΓB is connected, then G acts faithfully (Theorem 4.3.1(c))

on B as one of the 3-transitive permutation groups listed in Remark 4.4.1.

According to Theorem 5.1.2, under the assumption that D(B) contains no re-

peated blocks, all possibilities for the graphs Γ, ΓB, Γ[B,C] and the group G are

known if there are two adjacent vertices of Γ sharing the same second coordinate.

For the remaining case where the labels of any two adjacent vertices involve four

distinct blocks, the following theorem gives some structural information about Γ

and ΓB provided the girth of ΓB is sufficiently large.


invariant partition of V (Γ) with block size v = k + 1 ≥ 3 such that D(B) contains

no repeated blocks. Suppose further that girth(ΓB) ≥ 5. Then

(a) Γ[α, α′, β, β ′] ∼= K2 for adjacent blocks α, α′ and β, β ′ of P.

(b) Γ[B∗, C∗] is a matching for adjacent blocks B∗, C∗ of B∗, and if in addition

girth(ΓB) ≥ 7 then Γ[B∗, C∗] ∼= K2.

(c) The involution z : α 7→ α′ (α ∈ V (Γ)) defines a graph monomorphism from

Γ to the complement Γ, and z interchanges the two partitions B and B∗. Moreover,

z induces graph monomorphisms from ΓB to ΓB∗, and from ΓB∗ to ΓB, defined by

B 7→ B∗, and B∗ 7→ B, respectively.

Proof The assumption girth(ΓB) ≥ 5 implies that adjacent vertices of Γ have labels

involving four distinct blocks. Suppose that “BD”, “DB” and “CE”, “EC” are

blocks of P with “DB” and “EC” adjacent in Γ. (This is represented diagramat-

ically in Figure 3, where the two dashed boxes represent B∗ and C∗ respectively.)

Then B,C,D,E are pairwise distinct blocks by our assumption about the labels.

Note that “BD” is not adjacent to “EC” and “DB” is not adjacent to “CE” for

otherwise (B,D,E,B) or (C,D,E,C) would be a triangle of ΓB, contradicting the

assumption that girth(ΓB) ≥ 5. Similarly, “BD” = “DB”z is not adjacent to

“CE” = “EC”z, for otherwise (B,D,E,C,B) would be a 4-cycle of ΓB. Thus,

Γ[“BD”, “DB”, “CE”, “EC”] ∼= K2 and (a) holds.


In particular, the non-adjacency of “BD” and “CE” implies that z is a graph

monomorphism from Γ to Γ. By the definition of z, two vertices α, β lie in the same

block B of B if and only if αz, βz lie in the same block B∗ of B∗. Hence z induces

the bijection B 7→ B∗ from B to B∗. Suppose B∗, C∗ are adjacent blocks of B∗, say

“DB”, “EC” are adjacent vertices of Γ, where D ∈ ΓB(B), E ∈ ΓB(C) (see Figure

3). If B and C were adjacent in ΓB then (B,D,E,C,B) would be a 4-cycle in ΓB,

which is not the case. Thus B, C are not adjacent in ΓB, that is to say, if B, C

are adjacent in ΓB, then B∗, C∗ are not adjacent in ΓB∗ . Therefore, the bijection

B 7→ B∗ induced by z is a graph monomorphism from ΓB to ΓB∗ , and similarly the

bijection B∗ 7→ B is a graph monomorphism from ΓB∗ to ΓB.

If “DB” were adjacent to a second vertex, say “E1C”, in C∗, then (D,E,C,E1, D)

would be a 4-cycle of ΓB, contradicting the assumption that girth(ΓB) ≥ 5. There-

fore, Γ[B∗, C∗] is a matching. Now suppose girth(ΓB) ≥ 7, and suppose that there

is an edge “D1B”, “E1C” connecting B∗ and C∗, distinct from “DB”, “EC”.

If D1 = D then E1 6= E and (D,E,C,E1, D) is a 4-cycle, and similarly if E1 = E

then D1 6= D and (E,D1, B,D,E) is a 4-cycle. Hence D,E ∩ D1, E1 = ∅, but

in this case (B,D,E,C,E1, D1, B) is a 6-cycle. Hence Γ[B∗, C∗] ∼= K2. 2

r

r

pppppp

u

u

D

ppp

e

e

E

ppp

r

r

pppppp

B C

Figure 3 Blocks of B, B∗ and P

It is worth noticing that, under the assumptions of Theorem 5.1.3, the G-

invariant partition P satisfies all the assumptions of Section 4.2. Thus, from Theo-

rem 4.2.1, we know that the graphs Γ⋆ = Γ ∪ Γ′ and Γ# defined in Definition 4.2.1

with respect to P are both covers of ΓP .

Remark 5.1.2 From the group theoretic point of view (see, for example, [70, The-

orem 2.1(b)]), Theorem 5.1.3(c) shows that z carries the arc set Arc(Γ) of Γ to a

Three-arc Graphs 57

self-paired G-orbital on V (Γ) disjoint from Γ1 and hence z(Arc(Γ)) ⊆ Γi for some

i ≥ 2, where Γi := (α, β) : dΓ(α, β) = i. This parameter i might have a strong

influence on the structure of Γ. Essentially the same argument as that used in the

proof of Theorem 5.1.3 shows that i ≥ girth(ΓB) − 3 (so in particular i ≥ 2 if

girth(ΓB) ≥ 5). However, we have been unable to determine the exact value of i.

One consequence of Theorem 5.1.3 is that the valencies of Γ and ΓB∗ are bounded

as shown below. Recall that val(Γ) denotes the valency of a graph Γ.

Corollary 5.1.2 Under the assumptions of Theorem 5.1.3, val(Γ) ≤ (|V (Γ)|−2)/4,

and ΓB∗ has valency at most (|V (Γ)|/v) − v − 1. If in addition girth(ΓB) ≥ 7, then

val(Γ) ≤ (|V (Γ)|/v2) − (1/v) − 1.

Proof By Theorem 5.1.3, each edge of Γ joining α and β corresponds to a unique

3-path α, β ′, α′, β of Γ, and conversely each 3-path of Γ of this form corresponds to

a unique edge of Γ. One can see that the 3-paths of Γ with this form corresponding

to distinct edges of Γ are pairwise edge-disjoint, and that they have no common

edges with Γ′ (the latter being contained in Γ). So |E(Γ)| ≥ 3|E(Γ)| + |V (Γ)|/2,

that is, val(Γ) ≥ 3 · val(Γ) + 1. Thus, we have val(Γ) ≤ (|V (Γ)| − 2)/4. Now by

Theorem 5.1.3(c), we have val(ΓB)+val(ΓB∗) ≤ |B|−1 = (|V (Γ)|/v)−1, which yields

the second inequality since val(ΓB) = v by Theorem 5.1.1. Note that by Theorem

5.1.3(b), val(ΓB∗) = v · val(Γ) if girth(ΓB) ≥ 7, which implies the last inequality. 2

5.2 The 3-arc graph construction

As mentioned in Section 3.2, a fundamental problem arising from the geometric ap-

proach used in the thesis is that of reconstructing Γ from the triple (ΓB,Γ[B,C],D(B)).

In this section we study this problem for the case where k = v − 1 ≥ 2 and D(B)

contains no repeated blocks. In this case ΓB is (G, 2)-arc transitive by Theorem

5.1.2, and we will show that the reconstruction can be achieved satisfactorily. We

will give an explicit construction of such graphs from (G, 2)-arc transitive graphs

of valency v ≥ 3, and prove further that, up to isomorphism, it gives rise to all

G-symmetric graphs Γ with properties above.


We present the construction in a general setting, starting with a regular graph Σ

of valency v ≥ 3. For a subset ∆ of the set Arci(Σ) of i-arcs of Σ, the paired subset

of ∆ is defined by

∆ := (σi, σi−1, . . . , σ1, σ0) : (σ0, σ1, . . . , σi−1, σi) ∈ ∆

and ∆ is said to be self-paired if ∆ = ∆. The data needed for our construction are

a regular graph Σ and a self-paired subset of Arc3(Σ).

Definition 5.2.1 Let Σ be a regular graph of valency v ≥ 3, and let ∆ be a non-

empty self-paired subset of Arc3(Σ). Define Ξ(Σ,∆) to be the graph with vertex set

Arc(Σ) such that (σ, τ), (σ′, τ ′) ∈ Arc(Σ) are joined by an edge in Ξ(Σ,∆) if and

only if (τ, σ, σ′, τ ′) ∈ ∆. We call Ξ(Σ,∆) the 3-arc graph of Σ with respect to ∆.

The requirement that ∆ is self-paired ensures that adjacency in Ξ(Σ,∆) is well-

defined (in the sense that (σ, τ) is joined to (σ′, τ ′) if and only if (σ′, τ ′) is joined to

(σ, τ)). There are several natural partitions of the vertex set of Ξ(Σ,∆), namely

(i) P(Σ) := (σ, τ), (τ, σ) : (σ, τ) ∈ Arc(Σ);

(ii) B(Σ) := B(σ) : σ ∈ V (Σ), where B(σ) := (σ, τ) : τ ∈ Σ(σ);

(iii) B∗(Σ) := B∗(σ) : σ ∈ V (Σ), where B∗(σ) := (τ, σ) : τ ∈ Σ(σ).

Now let G be a group of automorphisms of Σ. Then G induces natural actions

on Arc(Σ) and Arc3(Σ), and provided G leaves ∆ invariant, G will preserve the

adjacency relation for Ξ(Σ,∆) and hence will induce an action as a group of auto-

morphisms of Ξ(Σ,∆). Moreover, the three partitions P(Σ), B(Σ) and B∗(Σ) are

all G-invariant. We note the following relations between the G-actions on Σ and

Ξ(Σ,∆): the proofs are straightforward and are omitted.

Lemma 5.2.1 Let Σ, ∆ be as in Definition 5.2.1, and let G be a group of automor-

phisms of Σ which leaves ∆ invariant. Then

(a) If G is faithful on the vertices of Σ, then it is also faithful on the vertices of

Ξ(Σ,∆).

(b) Ξ(Σ,∆) is G-vertex-transitive if and only if Σ is G-symmetric.

Three-arc Graphs 59

(c) Ξ(Σ,∆) is G-symmetric if and only if Σ is G-symmetric and ∆ is a self-paired

G-orbit on Arc3(Σ).

(d) For σ ∈ V (Σ), Gσ = GB(σ) = GB∗(σ), and the actions of Gσ on Σ(σ), B(σ)

and B∗(σ) are permutationally equivalent.

Thus, if Σ isG-symmetric and ∆ is a self-pairedG-orbit on Arc3(Σ), then Ξ(Σ,∆)

is an imprimitive G-symmetric graph relative to each of the partitions above. The

following self-evident lemma tells us when a G-orbit on Arc3(Σ) is self-paired, and

this will be used in the next two chapters.

Lemma 5.2.2 Suppose Σ is a G-symmetric graph. Then a G-orbit ∆ = (τ, σ, σ′, τ ′)G

on Arc3(Σ) is self-paired if and only if there exists an element of G which reverses

the 3-arc (τ, σ, σ′, τ ′), and this in turn is true if and only if there exists an element

of G which interchanges the arcs (σ, τ) and (σ′, τ ′).

Bearing in mind the remarks at the beginning of this section, we now study

3-arc graphs Ξ(Σ,∆) of a (G, 2)-arc transitive graph Σ with respect to self-paired

G-orbits ∆ on Arc3(Σ), paying particular attention to the partition B(Σ). A 3-arc

(τ, σ, σ′, τ ′) of Σ is said to be proper if τ 6= τ ′, that is, (τ, σ, σ′, τ ′) is not a 3-cycle.

Theorem 5.2.1 Suppose Σ is a (G, 2)-arc transitive graph with valency v ≥ 3.

Suppose ∆ is a self-paired G-orbit of 3-arcs of Σ. Set Γ := Ξ(Σ,∆). Then the

following (a)-(d) hold.

(a) For adjacent blocks B(σ), B(σ′) of ΓB(Σ), (σ, σ′) is the unique element of B(σ)

which is not adjacent to an element of B(σ′) (that is, “k = v− 1”), and the valency

of Γ[B(σ), B(σ′)] is equal to the size |(τ ′)Gτσσ′ | of the (Gτσσ′)-orbit containing τ ′,

where (τ, σ, σ′, τ ′) ∈ ∆. Hence val(Γ) = (val(Σ) − 1) · |(τ ′)Gτσσ′ |.

(b) ΓB(Σ)∼= Σ, and D(B(σ)) has no repeated blocks.

(c) If ∆ contains a 3-cycle then ∆ consists of all the 3-cycles of Σ, and both

Ξ(Σ,∆) and Σ are vertex disjoint unions of complete graphs, as specified in Theo-

rem 5.1.2 (b). The connected components of Ξ(Σ,∆) are the induced subgraphs on

the blocks of B∗(Σ).

(d) On the other hand if ∆ consists of proper 3-arcs then adjacent vertices of

Ξ(Σ,∆) involve four distinct vertices of Σ.


Proof Since B(σ), B(σ′) are adjacent in ΓB(Σ), there exist (σ, τ), (σ′, τ ′) ∈ Arc(Σ)

such that (τ, σ, σ′, τ ′) ∈ ∆. In particular (σ, σ′) ∈ Arc(Σ). Conversely, if (σ, σ′) ∈

Arc(Σ) then, since ∆ 6= ∅ and Σ is (G, 2)-arc transitive it follows that there exist

τ, τ ′ such that (τ, σ, σ′, τ ′) ∈ ∆ and hence such that (σ, τ) and (σ′, τ ′) are adjacent

in Γ. Thus B(σ) is adjacent to B(σ′) in ΓB(Σ). This proves that ΓB(Σ)∼= Σ.

It follows from the definition of a 3-arc that (σ, σ′) is not adjacent to any vertex

of B(σ′). Let (σ, ε) ∈ B(σ) with ε 6= σ′. Then some g ∈ G maps the 2-arc

(τ, σ, σ′) to the 2-arc (ε, σ, σ′) of Σ, and hence g maps the edge (σ, τ), (σ′, τ ′) of

Γ to (σ, ε), (σ′, (τ ′)g). Thus (σ, ε) is joined to some vertex of B(σ′) \ (σ′, σ).

It is now clear that the set of points of D(B(σ)) incident with the block B(σ′)

is B(σ) \ (σ, σ′). So D(B(σ)) has no repeated blocks. Clearly the valency of

Γ[B(σ), B(σ′)] is equal to |(τ ′)Gτσσ′ |, and from this the equality regarding the valency

of Γ follows.

If ∆ contains a 3-cycle then, since Σ is (G, 2)-arc transitive, the end vertices

of every 2-arc of Σ are adjacent vertices of Σ, so Σ is a disjoint union of complete

graphs. From the previous paragraph it follows that ∆ contains all the 3-cycles

of Σ, and that (σ, τ) is adjacent to (σ′, τ ′) in Ξ(Σ,∆) if and only if (σ, σ′) is an

arc of Σ and τ = τ ′. Thus the connected components of Ξ(Σ,∆) are the blocks

B∗(τ) of B∗(Σ) and each is a complete graph. By Lemma 5.2.1, the conditions of

Theorem 5.1.2 (b) hold, so Γ ∼= Ξ(Σ,∆) and ΓB(Σ)∼= Σ are as given there. On the

other hand, if ∆ consists of proper 3-arcs then adjacent vertices (σ, τ) and (σ′, τ ′)

of Ξ(Σ,∆) involve four distinct vertices of Σ. 2

Thus, under the assumptions of Theorem 5.2.1, we see that the graph Ξ(Σ,∆) is a

G-symmetric graph admitting the G-invariant partition B(Σ) such that k = v−1 ≥ 2

and D(B(σ)) contains no repeated blocks. We now show that every G-symmetric

graph Γ with these properties for some G-invariant partition B is isomorphic to a

3-arc graph Ξ(ΓB,∆) of the quotient graph ΓB, for a certain self-paired G-orbit ∆

on Arc3(ΓB). Note that in this case we may identify B with B(ΓB), and similarly

identify B∗,P defined in Theorem 5.1.1 with B∗(ΓB),P(ΓB) respectively.


invariant partition B of block size v = k+1 ≥ 3 such that D(B) contains no repeated

blocks, so ΓB is (G, 2)-arc transitive and the vertices of Γ are labelled with the arcs of

Three-arc Graphs 61

ΓB. Then Γ ∼= Ξ(ΓB,∆) for ∆ the (self-paired) G-orbit in Arc3(ΓB) containing the

3-arc (C,B,D,E), where (“BC”, “DE”) is an arc of Γ. In particular, ∆ contains

a 3-cycle if and only if Γ,ΓB are as in Theorem 5.1.2 (b).

Proof Let (“BC”, “DE”) be an arc of Γ. Then by the labelling defined before

Theorem 5.1.1, it is clear that (C,B,D,E) is a 3-arc of ΓB. Let ∆ be the G-

orbit containing it. Since G is transitive on Arc(Γ), ∆ is independent of the choice

of arc (“BC”, “DE”), and ∆ is self-paired. Since every arc of Γ is of the form

(“BgCg”, “DgEg”) for some g ∈ G, and since (Cg, Bg, Dg, Eg) = (C,B,D,E)g ∈ ∆,

it follows from Definition 5.2.1 that Γ ∼= Ξ(ΓB,∆). Finally, by Theorem 5.2.1 (c)

and (d), ∆ contains a 3-cycle if and only if the second coordinates of labels for

adjacent vertices of Γ are equal, and hence Γ,ΓB are as in Theorem 5.1.2 (b). 2

Remark 5.2.1 (a) The structure of Ξ(Σ,∆) for (G, 2)-arc transitive graphs Σ is

of considerable interest. We will explore in Chapter 7 the family of these graphs

for which Σ is a near-polygonal graph and ∆ is the set of 3-arcs occurring in the

distinguished “polygons” of Σ. This case is of particular interest in connection with

Section 5 of [43].

(b) The construction of the graphs Ξ(Σ,∆) bears some similarity to the cover-

ing graph construction of Biggs [6, pp.149-154]. The graphs Ξ(Σ,∆) are “almost

multicovers” of the 2-arc transitive graph Σ.

(c) Let Σ be a (G, 2)-arc transitive graph, and let σ, σ′ be a pair of adjacent

vertices of Σ. Then G contains an element g which interchanges σ and σ′. Let

τ ∈ Σ(σ) \ σ′. Then τ ′ := τ g ∈ Σ(σ′) \ σ, and (τ, σ, σ′, τ ′) is a 3-arc of Σ.

Also τ g2∈ Σ(σ) \ σ′. If it is possible to choose g and τ such that τ g

2= τ , then

g maps the 3-arc (τ, σ, σ′, τ ′) to its reverse (τ ′, σ′, σ, τ), and hence the G-orbit ∆

containing (τ, σ, σ′, τ ′) is self-paired. This is certainly possible if any one of the

following conditions holds:

(i) σ and σ′ are interchanged by an involution g;

(ii) the valency |Σ(σ)| of Σ is even (since we may take g to be a 2-element, and

g2 ∈ Gσσ′);

(iii) Σ is (G, 3)-arc transitive;


(iv) the actions of Gσσ′ on Σ(σ) \ σ′ and Σ(σ′) \ σ are permutationally iso-

morphic, in the sense that Gσσ′τ fixes a point ε ∈ Σ(σ′) \ σ, and σ′, τ are

the only points of Σ(σ) fixed by Gσσ′τ . (For if h ∈ Gσσ′ maps τ ′ to ε, then gh

interchanges σ and σ′, and maps τ to ε, and hence normalises Gσσ′τ = Gσσ′ε.

Therefore gh interchanges τ and ε, and hence reverses the 3-arc (τ, σ, σ′, ε).)

If any of these conditions holds, then Σ will occur as the quotient graph ΓB for a

graph Γ satisfying the hypotheses of Theorem 5.2.2.

To facilitate our later references, we summarize in the following the key infor-

mation contained in Theorems 5.1.2, 5.2.1 and 5.2.2.

Theorem 5.2.3 Let Γ be a G-symmetric graph, and B a nontrivial G-invariant

partition of V (Γ) with block size v ≥ 3 such that D(B) has block size v − 1. Then

D(B) contains no repeated blocks if and only if ΓB is (G, 2)-arc transitive. In this

case Γ ∼= Ξ(ΓB,∆) for some self-paired G-orbit ∆ of 3-arcs of ΓB. Conversely,

for any self-paired G-orbit ∆ of 3-arcs of a (G, 2)-arc transitive graph Σ of valency

v ≥ 3, the graph Γ = Ξ(Σ,∆), group G, and partition B(Σ) satisfy all the conditions

above.

5.3 Three-arc transitive quotient

From the discussion in the previous two sections, we see that even under the as-

sumption that k = v − 1 ≥ 2 and D(B) contains no repeated blocks, we are unable

to determine the graph Γ completely. This suggests that more information on either

the quotient graph ΓB or the bipartite graph Γ[B,C] may be needed in order to de-

termine Γ. With regard to the quotient, since we have proved that ΓB is (G, 2)-arc

transitive, we know that GB is 2-transitive on ΓB(B) by Lemma 3.1.1(b), and thus

we may naturally investigate the following two extreme cases:

(i) ΓB is (G, 3)-arc transitive;

(ii) GB is sharply 2-transitive on ΓB(B).

With respect to the bipartite graph Γ[B,C], we have also the following two extreme

cases in which Γ[B,C] contains the maximum and minimum possible numbers of

edges, respectively:

Three-arc Transitive Quotient 63

(I) Γ[B,C] ∼= Kv−1,v−1;

(II) Γ[B,C] ∼= (v − 1) ·K2.

The purpose of this section is to study the extreme case (I). We find (see Theorem

5.3.1 below) with surprise that case (I) occurs if and only if the extreme case (i)

for ΓB occurs, which in turn occurs if and only if the self-paired G-orbit ∆ needed

in Theorem 5.2.2 for reconstructing Γ is equal to Arc3(ΓB). Therefore, in this case

the graph Γ is uniquely determined by ΓB. In Chapter 7 we will study in detail the

extreme case (II), and in particular we will prove (Proposition 7.1.1) that (ii) occurs

only if (II) occurs.


invariant partition of V (Γ) with block size v = k + 1 ≥ 3 such that D(B) contains

no repeated blocks. Then the following conditions (a)-(c) are equivalent:

(a) ΓB is (G, 3)-arc transitive;

(b) Γ[B,C] ∼= Kv−1,v−1;

(c) Γ ∼= Ξ(ΓB,∆) with ∆ the set of all 3-arcs of ΓB.

Thus in this case Γ is uniquely determined by ΓB.

Proof Since D(B) has no repeated blocks, ΓB is (G, 2)-arc transitive by Theo-

rem 5.1.2. Suppose that (“BC”, “DE”) is an arc of Γ and let ∆ be the G-orbit on

Arc3(ΓB) containing the 3-arc (C,B,D,E). By Theorem 5.2.2, Γ ∼= Ξ(ΓB,∆). Now

each 3-arc (C1, B,D,E1) of ΓB corresponds to a unique ordered pair “BC1”, “DE1”

of vertices of Γ and vice versa, where C1 ∈ ΓB(B) \ D and E1 ∈ ΓB(D) \

B. Thus we have the following: Γ[B,D] ∼= Kv−1,v−1 ⇔ for any such C1, E1,

“BC1”, “DE1” are adjacent in Γ ⇔ for any such C1, E1, there exists g ∈ G with

(“BC”, “DE”)g = (“BC1”, “DE1”) ⇔ for any such C1, E1, there exists g ∈ G with

(C,B,D,E)g = (C1, B,D,E1) ⇔ for any such C1, E1, the 3-arc (C1, B,D,E1) is

in ∆ ⇔ ∆ = Arc3(ΓB) ⇔ ΓB is (G, 3)-arc transitive. Thus (a), (b) and (c) are

equivalent. 2

Chapter 6

Three-arc graphs of completegraphs: Classification

When things had been classified in organic categories, knowledge

moved toward fulfillness.


The purpose of this chapter is to classify all G-symmetric graphs which admit a

G-invariant partition B such that k = v − 1 ≥ 2, D(B) contains no repeated blocks

and ΓB∼= Kv+1 (note that val(ΓB) = v by Theorem 5.1.1), where G ≤ Aut(Γ).

By Theorem 5.2.3, this is equivalent to classifying 3-arc graphs of complete (G, 2)-

arc transitive graphs Σ := Kv+1. In this case G must be 3-transitive on V (Σ) (see

Lemma 6.1.1 below), and by Theorem 4.3.1(c) and Lemma 5.2.1(a), G is also faithful

on V (Σ). Hence, by the classification of highly transitive groups (see Theorem 2.1.1

and the comments following it), G is one of the following groups of degree v+1 with

the natural 3-transitive permutation representation on V (Σ):

(i) Sv+1 (v ≥ 3);

(ii) Av+1 (v ≥ 4);

(iii) AGL(d, 2) (v = 2d − 1 ≥ 3);

(iv) Z42.A7 (v = 15);

(v) Mathieu groups Mv+1 (v = 10, 11, 21, 22, 23) and M11 (v = 11); and

66 Classification

(vi) 3-transitive groups G satisfying PGL(2, v) ≤ G ≤ PΓL(2, v) (v ≥ 3 is a prime

power).

We will classify all the 3-arc graphs of Σ with respect to self-paired G-orbits

on Arc3(Σ). The feasibility of such a classification is due to the classification of 3-

transitive permutation groups, as shown above, and hence relies on the classification

of finite simple groups. To study 3-arc graphs of Σ arising from the groups G in (vi),

we need a detailed description of the 3-transitive subgroups of PΓL(2, v), and this

will be given in Section 6.2. The 3-arc graphs obtained in this case are the so-called

cross-ratio graphs, which were first introduced in [43] and studied systematically in

[46]. Those arising from the other 3-transitive groups were classified in [45]. The

3-arc graphs arising from the groups G in (iii) and (iv) belong to a large class of

symmetric graphs associated with the classical affine geometries which we will study

in detail in Section 9.5. The results obtained in this chapter will be used in the next

chapter.

6.1 Simple examples

Lemma 6.1.1 Let Σ be a connected (G, 2)-arc transitive graph with valency v ≥ 3.

Then girth(Σ) = 3 if and only if Σ ∼= Kv+1, which in turn is true if and only if G is

3-transitive on V (Σ).

Proof If Σ ∼= Kv+1, then girth(Σ) = 3 and G is 3-transitive on V (Σ) since Gσ is

2-transitive on Σ(σ) = V (Σ) \ σ and G is transitive on V (Σ). Next suppose that

G is 3-transitive on V (Σ). Then, for each σ ∈ V (Σ), Gσ is 2-transitive on V (Σ)\σ

and hence V (Σ) \ σ induces a complete graph Kv (note that V (Σ) \ σ contains

adjacent vertices). This implies Σ ∼= Kv+1. Finally, if girth(Σ) = 3, then Σ(σ)

induces a complete graph Kv by the 2-transitivity of Gσ on Σ(σ). Hence Σ ∼= Kv+1

by the connectedness of Σ. 2

In the remaining part of this chapter we will suppose Σ := Kv+1, G is one of the

groups in (i)-(vi), and Γ := Ξ(Σ,∆) with ∆ a self-paired G-orbit on Arc3(Σ). For

simplicity we use στ to denote the arc (σ, τ) of Σ if there is no danger of confusion.

We begin with the following two simple examples.

Simple Examples 67

Example 6.1.1 Unions of complete graphs. If ∆ contains a 3-cycle of Σ, then

by Theorem 5.2.1(c), ∆ is the set of all 3-cycles of V (Σ), and in this case we have

Γ = Ξ(Σ,∆) ∼= (v+1) ·Kv, Γ[B(σ), B(σ′)] ∼= (v−1) ·K2 for any two distinct vertices

σ, σ′ of Σ, and G can be any one of the groups listed in (i)-(vi) above.

Therefore, in the following discussion we may suppose that ∆ consists of proper

3-arcs of Σ. Also by the 3-transitivity of G on V (Σ), to seek self-paired G-orbits

∆ := (σ′, σ, τ, τ ′)G on Arc3(Σ), we can start from any chosen 2-arc (σ′, σ, τ) of Σ.

The next example determines all the 3-arc graphs of Σ (other than (v + 1) · Kv)

arising from 4-transitive groups. For integers ℓ, n with 2 ≤ 2ℓ < n, the Kneser

graph K(n, ℓ) is the graph with vertices all ℓ-subsets of a given n-set in which two

such ℓ-subsets X, Y are adjacent if and only if X ∩ Y = ∅.

Example 6.1.2 Let Σ := Kv+1, and let G be 4-transitive on V (Σ). Then either

G = Sv+1 (v ≥ 3), or G = Av+1 (v ≥ 5), or G = Mv+1 (v = 10, 11, 22, 23).

In each case, G is transitive on the set ∆ of all proper 3-arcs of Σ, and hence

∆ is the unique self-paired G-orbit on such 3-arcs. As mentioned in Section 5.2,

P := στ, τσ : σ, τ ∈ V (Σ), σ 6= τ is a G-invariant partition of the vertex set of

Γ = Ξ(Σ,∆). One can see that two blocks P := στ, τσ, Q := δε, εδ of P are

adjacent if and only if σ, τ ∩ δ, ε = ∅, and in this case we have Γ[P,Q] ∼= K2,2.

So ΓP is isomorphic to the Kneser graph K(v + 1, 2), and στ, δε are adjacent in Γ

if and only if σ, τ ∩ δ, ε = ∅. Thus Γ is isomorphic to (K(v + 1, 2))[K2], the

lexicographic product of K(v + 1, 2) by the empty graph K2 on two vertices. One

can see that, for distinct blocks B,C of B(Σ), Γ[B,C] is isomorphic to Kv−1,v−1

minus a perfect matching.

So the only remaining groups are those in (iii), (iv), (vi), and the Mathieu

groups M11 (with degree v + 1 = 12) and M22 (with degree v + 1 = 22). (Note that

S4∼= PGL(2, 3) and A5

∼= PGL(2, 4).) In the remaining sections of this chapter we

will study the 3-arc graphs arising from such groups.

68 Classification

6.2 The 3-transitive subgroups of PΓL(2, v)

Let v = pe where p is a prime and e ≥ 1. Then the multiplicative group GF(v)# of

units of the finite field GF(v) is a cyclic group of order pe − 1. As is well-known,

Sq(v) := x2 : x ∈ GF(v)#

is a subgroup of GF(v)# with index one or two according as v is even or odd. We

may identify the projective line PG(1, v) with GF(v) ∪ ∞ by identifying a point

[(y, z)] of the former with the element y/z of the latter, where ∞ satisfies the usual

arithmetic rules such as 1/∞ = 0, ∞−y = ∞, y−∞ = −∞, (∞· y)/(∞· z) = y/z,

∞p = ∞, etc. With respect to the bijection [(y, z)] 7→ y/z, the action of PGL(2, v)

on the points of PG(1, v) is permutationally equivalent (see e.g. [10, 63]) to the

action of the group of Mobius transformations

ta,b,c,d : z 7→az + b

cz + d(a, b, c, d ∈ GF(v), ad− bc 6= 0)

acting on GF(v) ∪ ∞. So we may identify PGL(2, v) with this group in the

following. Then PSL(2, v) = ta,b,c,d : ad − bc ∈ Sq(v), and it follows from the

definition that PΓL(2, v) = PGL(2, v).〈ψ〉, the semidirect product of PGL(2, v) by

〈ψ〉, where ψ is the Frobenius mapping defined by

ψ : z 7→ zp, z ∈ GF(v) ∪ ∞. (6.1)

For an integer i with 0 ≤ i < e, we call (i, ta,b,c,d) a twisted pair if either i is even and

ad − bc ∈ Sq(v), or i is odd and ad − bc ∈ GF(v)# \ Sq(v). The following theorem

shows that the 3-transitive subgroups of PΓL(2, v) fall into two categories.

Theorem 6.2.1 Let v = pe, where p is a prime and e ≥ 1. Then a group G with

PSL(2, v) ≤ G ≤ PΓL(2, v) is 3-transitive on GF(v) ∪ ∞ if and only if G is one

of the following:

(a) G = PGL(2, v).〈ψn〉 for n a divisor of e;

(b) G = M(n, v) := ψinta,b,c,d : (i, ta,b,c,d) a twisted pair, where p is an odd

prime, e ≥ 2 is an even integer and n is a divisor of e/2.

Proof Suppose first that PGL(2, v) ≤ G ≤ PΓL(2, v). Then, since PGL(2, v) is

(sharply) 3-transitive on GF(v) ∪ ∞ (e.g. [10, Theorem 2.6.2]) and PΓL(2, v) =

PGL(2, v).〈ψ〉, we have G = PGL(2, v).〈ψn〉 for some divisor n of e.

Three-transitive Subgroups of PΓL(2, q) 69

In the following we suppose that PSL(2, v) < G < PΓL(2, v), G 6≥ PGL(2, v)

and that G is 3-transitive on GF(v) ∪ ∞. Since PSL(2, 2e) = PGL(2, 2e), p must

be an odd prime (and hence Sq(v) has index two in GF(v)#). Let θ be a fixed

element of PGL(2, v) which is not in PSL(2, v), and let ψ, θ be the left cosets

of PSL(2, v) containing ψ, θ respectively. Then PGL(2, v)/PSL(2, v) = 〈θ〉 ∼= Z2

and PΓL(2, v)/PSL(2, v) = 〈ψ〉 × 〈θ〉 ∼= Ze × Z2. Since PGL(2, v) 6≤ G, we have

G := G/PSL(2, v) = 〈ψnθt〉 for a divisor n of e with 1 < n < e and t = 0 or 1. If

t = 0, then G = PSL(2, v).〈ψn〉 and hence G0∞ = ψinta,0,0,1 : 0 ≤ i < e, a ∈ Sq(v).

Thus G is not 3-transitive on GF(v) ∪ ∞ since Sq(v) 6= GF(v)# and since each

element ψinta,0,0,1 in G0∞ maps 1 to a ∈ Sq(v). This contradiction shows that t = 1

and hence G = 〈ψnθ〉. If e/n is odd, then (ψnθ)e/n = θe/n = θ ∈ G, which is not the

case as PGL(2, v) 6≤ G. Hence e is even and n divides e/2. Note that (i, θita,b,c,d) is

a twisted pair for each i with 0 ≤ i < e/n and for any ta,b,c,d ∈ PSL(2, v). Therefore,

we have G = ψinθita,b,c,d : 0 ≤ i < e/n, ad− bc ∈ Sq(v) = M(n, v).

To complete the proof, one can see that M(n, v) is 3-transitive on GF(v) ∪ ∞

for any v = pe with p an odd prime and e ≥ 2 an even integer and for any divisor n

of e/2. 2

Note that if n = e then PGL(2, v).〈ψn〉 = PGL(2, v). For p, e, n as in part (b) of

Theorem 6.2.1, it follows from the definition that M(n, v) = 〈PSL(2, v), ψnta,0,0,1〉,

where a is a primitive element of GF(v). This expression of M(n, v) is independent

of the choice of the element a.

Corollary 6.2.1 Let v = pe, where p is a prime and e ≥ 1. Let G be a 3-transitive

subgroup of PΓL(2, v), as specified in Theorem 6.2.1. Then G∞01 = 〈ψn〉 if G =

PGL(2, v).〈ψn〉 (n a divisor of e); and G∞01 = 〈ψ2n〉 if G = M(n, v) (for suitable

p, e, n).

Proof For G = PGL(2, v) · 〈ψn〉, we have G∞0 = ψinta,0,0,1 : i ≥ 0, a ∈ GF(v)#.

So we get G∞01 = 〈ψn〉. On the other hand, for G = M(n, v), by definition we have

G∞01 = ψin : i is even = 〈ψ2n〉. 2

In particular, this implies that (M(e/2, v))∞01 = 1 and hence M(e/2, v) is sharply

3-transitive on GF(v) ∪ ∞ (see e.g. [26, pp. 242-243] for details on this group).

70 Classification

6.3 Definitions of cross-ratio graphs

In this section, we use Σ to denote the complete graph Kv+1 with vertex set GF(v)∪

∞, where v = pe with p a prime and e ≥ 1 an integer. So Σ has arc set

Ω(v) := yz : y, z ∈ GF(v) ∪ ∞, y 6= z.

Suppose G is a 3-transitive subgroup of PΓL(2, v), so that Σ is (G, 2)-arc transitive.

We will define cross-ratio graphs as 3-arc graphs of Σ with respect to self-paired G-

orbits on Arc3(Σ). In accordance with Theorem 6.2.1, we distinguish the following

two cases.

We first consider 3-arc graphs of Σ arising from 3-transitive groups G given in

Theorem 6.2.1(a). From the theory of finite fields, for each element x ∈ GF(v)\0,

the subfield of GF(v) generated by x has the form GF(pn(x)), for some divisor n(x)

of e.

Lemma 6.3.1 Let x ∈ GF(v) \ 0, 1. Let n be a divisor of n(x) and G :=

PGL(2, v).〈ψn〉. Then ∆ := (0,∞, 1, x)G is a self-paired G-orbit on Arc3(Σ).

Proof Since 1 · (−1)− (−x) ·1 = x−1 6= 0, t1,−x,1,−1 is an element of PGL(2, v). So

t1,−x,1,−1 is an element of G since PGL(2, v) ≤ G. Clearly, t1,−x,1,−1 maps (0,∞, 1, x)

to (x, 1,∞, 0). Hence, by Lemma 5.2.2, ∆ is self-paired. 2

Definition 6.3.1 Let x, n, G and ∆ be as in Lemma 6.3.1. Then the 3-arc graph

Ξ(Σ,∆) of Σ with respect to ∆ is well-defined. We call this graph an untwisted

cross-ratio graph and denote it by CR(v; x, n).

For 3-transitive subgroups of PΓL(2, v) given in Theorem 6.2.1(b), we have the

following lemma.

Lemma 6.3.2 Let v = pe with p an odd prime and e an even integer, and let

x ∈ GF(v) \ 0, 1 be such that n(x) is even and x − 1 ∈ Sq(v). Let n be an even

divisor of n(x) and let G := M(n/2, v). Then ∆ := (0,∞, 1, x)G is a self-paired

G-orbit on Arc3(Σ).

Cross-ratio Graphs 71

Proof Since x − 1 ∈ Sq(v), we have t1,−x,1,−1 ∈ PSL(2, v). Hence t1,−x,1,−1 ∈

M(n/2, v) as PSL(2, v) ≤ M(n/2, v). Since t1,−x,1,−1 reverses (0,∞, 1, x), the result

follows immediately from Lemma 5.2.2. 2

Definition 6.3.2 Let p, e, x, n, G and ∆ be as in Lemma 6.3.2. Then the 3-arc

graph Ξ(Σ,∆) of Σ with respect to ∆ is well-defined. We call this graph a twisted

cross-ratio graph and denote it by TCR(v; x, n).

In the following theorem, we show that the (untwisted and twisted) cross-ratio

graphs can be defined equivalently in terms of cross-ratios. This approach for defin-

ing untwisted cross-ratio graphs was adopted in [46], and it justifies the terminology

used. For distinct elements u, w, y, z ∈ GF(v) ∪ ∞, the cross-ratio is defined as

c(u, w; y, z) :=(u− y)(w − z)

(u− z)(w − y)

(see e.g. [63, pp. 59]) with the usual convention for ∞ as mentioned in previous

section. The cross-ratio can take all values in GF(v) except 0 and 1. For x ∈

GF(v) \ 0, 1 and a divisor n of n(x), the field automorphism ψn acts on GF(pn(x))

and there are exactly n(x)/n images of x under the elements of 〈ψn〉, namely

B(x, n) := xψin

: 0 ≤ i < n(x)/n. (6.2)

Thus B(x, n) is the 〈ψn〉-orbit on GF(pn(x)) containing x.

Theorem 6.3.1 Suppose v = pe, where p is a prime and e ≥ 1 an integer.

(a) Let x ∈ GF(v) \ 0, 1. Let n be a divisor of n(x) and G := PGL(2, v).〈ψn〉.

Then CR(v; x, n) is the G-symmetric graph with vertex set Ω(v) in which uw and

yz are adjacent if and only if c(u, w; y, z) ∈ B(x, n).

(b) Let x ∈ GF(v) \ 0, 1 be such that n(x) is even and x − 1 ∈ Sq(v). Let n

be an even divisor of n(x) and let G := M(n/2, v) (where p is odd and e is even).

Then TCR(v; x, n) is the G-symmetric graph with vertex set Ω(v) in which yz and

∞0 are adjacent if and only if y ∈ GF(v)# and

z ∈

B(x, n)y, if y ∈ Sq(v)

B(x, n)ψn/2y, if y ∈ GF(v)# \ Sq(v).

72 Classification

Proof The cross-ratio is invariant under the action of PGL(2, v) on 4-tuples of

distinct elements of GF(v) ∪ ∞, and moreover PGL(2, v) is transitive on such

4-tuples with a fixed cross-ratio (see e.g. [63, pp. 59]). Under the action of the

Frobenius mapping ψ, we have

c(uψ, wψ; yψ, zψ) = (c(u, w; y, z))ψ.

(a) By the definition of CR(v; x, n) as a 3-arc graph and by Lemma 5.2.1(c),

CR(v; x, n) is G-symmetric. Since c(∞, 0; 1, x) = x, by the definition of a 3-arc

graph we have: uw and yz are adjacent in CR(v; x, n) ⇔ (w, u, y, z) ∈ (0,∞, 1, x)G

⇔ c(u, w, y, z) ∈ B(x, n).

(b) This can be proved in a similar manner, by using the properties of the cross-

ratio mentioned above. 2

Since B(x, n(x)) = x, part (a) of Theorem 6.3.1 implies that two vertices uw

and yz are adjacent in CR(v; x, n(x)) if and only if c(u, w; y, z) = x. Recall that

B(x, n) is the 〈ψn〉-orbit containing x, so in part (b) of Theorem 6.3.1 the sets

B(x, n) and B(x, n)ψn/2

are disjoint and their union is B(x, n/2).

From the discussion in Section 5.2, for Γ = CR(v; x, n) and G = PGL(2, v).〈ψn〉,

or for Γ = TCR(v; x, n) and G = M(n/2, v), the vertices of Γ admit the following

three G-invariant partitions:

P(v) := yz, zy : y, z ∈ GF(v) ∪ ∞, y 6= z;

B(v) := B(y) : y ∈ GF(v) ∪ ∞, where B(y) := yz : z ∈ GF(v) ∪ ∞, y 6= z;

B∗(v) := B∗(y) : y ∈ GF(v)∪∞, where B∗(y) := zy : z ∈ GF(v)∪∞, y 6= z.

Moreover, yz is the unique vertex of B(y) not adjacent to any vertex of B(z) in Γ.

Since |B(x, n)| = n(x)/n and ΓB(v)∼= Kv+1 (Theorem 5.2.1(b)), this together with

Theorem 6.3.1 implies the following consequence.

Corollary 6.3.1 Let Γ = CR(v; x, n) or Γ = TCR(v; x, n) (for proper x and n).

Then for distinct blocks B,C of B(v), the graph Γ[B,C] has valency n(x)/n. Hence

the valency of Γ is equal to (q − 1)n(x)/n.

More results concerning cross-ratio graphs can be found in [46]. For example,

all instances of isomorphism between the cross-ratio graphs are determined in [46,

Characterizing Cross-ratio Graphs 73

Theorem 7.2]. In particular, there are no isomorphisms between an untwisted cross-

ratio graph and a twisted cross-ratio graph.

6.4 Characterizing cross-ratio graphs

In this section we will show that, for G a 3-transitive subgroup of PΓL(2, v), the

(twisted or untwisted) cross-ratio graphs are the only 3-arc graphs of Σ = Kv+1

with respect to self-paired G-orbits on proper 3-arcs of Σ. In fact, we have the

following characterization for the cross-ratio graphs. Here we adopt the notation in

the previous section.

Theorem 6.4.1 Let v = pe ≥ 3, where p is a prime and e ≥ 1. Suppose that Γ

is a G-symmetric graph with vertex set Ω(v), where G is a 3-transitive subgroup of

PΓL(2, v) with the induced natural action on Ω(v). Then either

(a) Γ ∼= (v + 1) ·Kv, with connected components being either the blocks of B(v)

or the blocks of B∗(v), or

(b) Γ ∼=(

v+12

)

·K2, with connected components the blocks of P(v), or

(c) Γ is isomorphic to CR(v; x, n) or TCR(v; x, n) for some x, n.

Proof Let (∞0, dx) be an arc of Γ. If d = ∞, then since G is 3-transitive on

GF(v) ∪ ∞ and since the action of G on Ω(v) is induced by the action of G on

GF(v) ∪ ∞, we know that two vertices uw, yz of Γ are adjacent if and only if

u = y, that is, if and only if (a) holds with components the blocks B(u) of B(v), for

u ∈ GF(v) ∪ ∞. Similarly, if x = 0 then (a) holds with components the blocks

B∗(u) of B∗(v), for u ∈ GF(v) ∪ ∞; and if dx = 0∞ then (b) holds. So suppose

in the following that d 6= ∞, x 6= 0, and dx 6= 0∞. Suppose that d = 0, so that

x 6= ∞. Any element of G which maps ∞0 to 0x must map 0x to xz, for some z,

and hence there is no element of G which interchanges ∞0 and 0x, contradicting the

arc-transitivity of Γ. Hence d 6= 0 and similarly x 6= ∞, so ∞, 0, d, x are pairwise

distinct. Since G is 3-transitive on GF(v) ∪ ∞, we may assume that d = 1.

By Theorem 6.2.1, for some divisor n of e, we have G = PGL(2, v).〈ψn〉, or G =

M(n/2, v), where in the latter case p is odd and both e and n are even. By Corollary

6.2.1, G∞01 = 〈ψn〉, and since G is 3-transitive on GF(v) ∪ ∞, and transitive on

arcs of Γ, it follows that 〈ψn〉 is transitive on the vertices of Γ(∞0)∩B(1). Thus this

74 Classification

set consists of all pairs 1x′, for x′ ∈ U(x) := xψin

: for some i. We can determine

Γ(∞0) since it is the orbit of G∞0 containing 1x. If G = PGL(2, v) · 〈ψn〉 then

Γ(∞0) consists of the pairs uw where w ∈ U(x)u. If G = M(n/2, v), then Γ(∞0)

consists of the pairs uw where w ∈ U(x)u if u is a square, and where w ∈ U(x)ψn/2u

if u is not a square.

The set U(x) is contained in the subfield GF(pn(x)) generated by x, and so each

element of U(x) is left invariant by ψn(x). Moreover ψn(x) maps squares to squares.

It follows that Γ(∞0) is left invariant by 〈G∞0, ψn(x)〉 and hence that 〈G,ψn(x)〉

leaves the set of arcs of Γ invariant, that is, 〈G,ψn(x)〉 is contained in Aut(Γ). Thus

we may assume that ψn(x) ∈ G, and hence that n divides n(x). This means that

U(x) is the set B(x, n), as defined in (6.2). If G = PGL(2, v) · 〈ψn〉 then we have

shown that the set of vertices adjacent to ∞0 is the same for Γ and CR(v; x, n), and

they admit the same arc-transitive group G. Hence in this case Γ = CR(v; x, n).

Suppose therefore that G = M(n/2, v). Since G is arc-transitive on Γ, some

element g = ψita,b,c,d of G interchanges ∞0 and 1x. Since g interchanges ∞ and

1, and maps 0 to x, we have g = ψit1,−x,1,−1. Then, since g maps x to 0, we have

xψi= x, and hence n(x) divides i. Since n divides n(x), this means that n divides

i, and hence ψi ∈ G. Therefore t1,−x,1,−1 ∈ G ∩ PGL(2, v) = PSL(2, v), and so

x − 1 ∈ Sq(v). Therefore the graph TCR(v; x, n) is defined, and we have shown

that the set of vertices adjacent to ∞0 is the same for Γ and TCR(v; x, n), and they

admit the same arc-transitive group M(n/2, v). Hence in this case Γ = TCR(v; x, n).

2

Theorem 6.4.1 and its proof imply the following corollary.

Corollary 6.4.1 Let v = pe ≥ 3 with p a prime and e ≥ 1.

(a) The graphs CR(v; x, n), TCR(v; x, n) and (v + 1) ·Kv (as in Example 6.1.1)

are the only 3-arc graphs of Σ = Kv+1 with respect to self-paired G-orbits on Arc3(Σ),

where G is a 3-transitive subgroup of PΓL(2, v).

(b) For Γ = CR(v; x, n) or Γ = TCR(v; x, n), the 3-transitive subgroup H such

that Γ is H-symmetric is equal to PGL(2, v) · 〈ψt〉 or M(t/2, v) respectively, for some

divisor t of e such that gcd(n(x), t) = n.

Affine Three-arc Graphs 75

6.5 Affine 3-arc graphs

For an integer d ≥ 2 and a prime power q, we use V (d, q) to denote the d-dimensional

linear space of row vectors over GF(q). We use ei to denote the unit vector of V (d, q)

with i-th coordinate 1 and the remaining coordinates 0, for i = 1, 2, . . . , d. The affine

group AGL(d, q) consists of all affine transformations

tM,w : z 7→ zM + w (6.3)

of V (d, q), whereM is a d×d invertible matrix over GF(q) and w ∈ V (d, q). Similarly

the group AΓL(d, q) consists all semilinear transformations tM,w,ρ : z 7→ zρM + w

of V (d, q), where ρ ∈ Aut(GF(q)) and ρ acts componentwise on vectors of V (d, q).

The affine geometry AG(d, q) is the geometry with point set V (d, q) and n-flats

(1 ≤ n ≤ d) of the form U + w := u + w : u ∈ U, where U is an n-dimensional

subspace of V (d, q) and w ∈ V (d, q). A 1-flat (2-flat, respectively) of AG(d, q) is

usually called a line (plane, respectively) of AG(d, q). Three points of AG(d, q) are

said to be collinear if they lie on a line, and four points of AG(d, q) are said to be

coplanar if they lie on a plane.

In studying 3-arc graphs of Σ = Kv+1 arising from the groups in (iii) and (iv),

we need the following basic result for AG(d, q), which we will use in Section 9.5 as

well.

Lemma 6.5.1 Suppose AGL(d, q) ≤ G ≤ AΓL(d, q), where d ≥ 2 and q is a prime

power. Then, for 1 ≤ n ≤ d, G is transitive on ordered (n + 1)-tuples of points of

AG(d, q) not lying on any (n− 1)-flat of AG(d, q).

Proof Any given n + 1 points x0,x1, . . . ,xn of AG(d, q) do not lie on the same

(n−1)-flat if and only if x1−x0, . . . ,xn−x0 are independent vectors of V (d, q). So in

this case x1−x0, . . . ,xn−x0 can be taken as the first n vectors of an ordered base of

V (d, q). Hence there exists tM,0 ∈ GL(d, q) which maps e1, . . . , en to x1−x0, . . . ,xn−

x0, respectively. Thus tM,x0 ∈ AGL(d, q) maps (0, e1, . . . , en) to (x0,x1, . . . ,xn).

Since (0, e1, . . . , en) is a typical (n+1)-tuple not lying on any (n−1)-flat of AG(d, q),

the result follows immediately. 2

We now determine the 3-arc graphs arising from the 3-transitive affine group (iii)

in the introduction of this chapter. These graphs were classified in [45].

76 Classification

Example 6.5.1 Affine 3-arc graphs. Let G := AGL(d, 2), d ≥ 2, and let Σ be the

complete graph with vertex set V (d, 2). Then the proper 3-arcs (τ, σ, σ′, τ ′) of Σ can

be partitioned into the following two parts:

∆1 := ∆1(d, 2) = (τ, σ, σ′, τ ′) : τ, σ, σ′, τ ′ coplanar in AG(d, 2),

∆2 := ∆2(d, 2) = (τ, σ, σ′, τ ′) : τ, σ, σ′, τ ′ non-coplanar in AG(d, 2).

Clearly, both ∆1 and ∆2 are self-paired. Note that each line of AG(d, 2) contains

exactly two points, and each plane of AG(d, 2) contains exactly four points (see

e.g. [84, Theorem 1.17]). Thus, for any proper 3-arc (τ, σ, σ′, τ ′) in ∆1, the points

τ, σ, σ′ are non-collinear and moreover we have τ − σ = τ ′ − σ′. This together

with Lemma 6.5.1 implies that G is transitive on ∆1. Similarly, G is transitive on

∆2. Since G preserves coplanarity, we conclude that ∆1, ∆2 are both self-paired

G-orbits on proper 3-arcs of Σ, and they are the only such G-orbits. So we get two

3-arc graphs of Σ, namely Ξi(d, 2) := Ξ(Σ,∆i) for i = 1, 2. (In defining the graph

Ξ2(d, 2) we require that d ≥ 3 since ∆2 6= ∅ if and only if d ≥ 3.) It follows from

the definition that Ξ1(d, 2) is the graph with vertices the ordered pairs of distinct

vectors of V (d, 2) in which uw,yz are adjacent if and only if u,w,y, z are distinct

and u− w = y − z. Also, Ξ2(d, 2) is the graph with the same vertices in which

uw,yz are adjacent if and only if u,w,y, z are non-coplanar in AG(d, 2).

Example 6.5.2 The group G := Z42.A7 is a subgroup of AGL(4, 2), where Z

42 acts

on V (Σ) := V (4, 2) by translations and, for τ := 0, Gτ∼= A7 is a subgroup of

GL(4, 2) ∼= A8 acting 2-transitively on V (4, 2) \ τ in its natural action. Let σ, σ′

be distinct points of V (4, 2) \ τ. Then from [24, pp.10] we have Gστ∼= PSL(2, 7),

which is transitive on V (4, 2) \ σ, τ, and each involution in A7 and also each

element of order 3 in PSL(2, 7) fixes exactly 3 nonzero vectors in V (4, 2). Hence in

the action of Gσσ′τ∼= A4 on V (4, 2) \ σ, σ′, τ, σ+σ′ + τ the stabilizer of any vector

is trivial, that is, Gσσ′τ has an orbit of length 12. Apart from this orbit, Gσσ′τ has

another orbit on V (4, 2) \ σ, σ′, τ, namely σ + σ′ + τ. Since G is 3-transitive

on V (Σ), there are two G-orbits on proper 3-arcs of Σ. It is clear that these two

G-orbits are ∆1(4, 2) and ∆2(4, 2). Therefore, we have exactly two 3-arc graphs of

Σ, namely Ξ1(4, 2) and Ξ2(4, 2).

Classification Theorem 77

6.6 Mathieu graphs, and the classification

theorem

In this last section we determine 3-arc graphs from the two Mathieu groups M11

(with degree v + 1 = 12) and M22 (with degree v + 1 = 22), and thus complete our

classification. These graphs were classified in [45].

Example 6.6.1 The Mathieu group M11 with degree v + 1 = 12 is the automor-

phism group of the unique 3-(12, 6, 2) design D. We assume that the point set

of D is the same as the vertex set of Σ := K12. For a 2-arc (σ′, σ, τ) of Σ,

let X(σ′, σ, τ) denote the union of the two blocks of D containing σ′, σ, τ . Then

(M11)σ′στ ∼= S3 has two orbits on V (Σ) \ σ′, σ, τ (see [26, pp.231-232]), namely

V (Σ) \ X(σ′, σ, τ) and X(σ′, σ, τ) \ σ′, σ, τ. Let τ ′ ∈ V (Σ) \ σ′, σ, τ. By the

3-transitivity of M11, there exists g ∈ M11 such that (σ, τ, τ ′)g = (τ, σ, σ′). Set

(σ′)g = δ, so (σ′, σ, τ, τ ′)g = (δ, τ, σ, σ′). Since g is an automorphism of D, the

points σ′, σ, τ, τ ′ lie in the same block of D if and only if δ, τ, σ, σ′ lie in the same

block of D. This implies that, τ ′ ∈ V (Σ) \X(σ′, σ, τ) (τ ′ ∈ X(σ′, σ, τ) \ σ′, σ, τ,

respectively) if and only if δ ∈ V (Σ) \ X(σ′, σ, τ) (δ ∈ X(σ′, σ, τ) \ σ′, σ, τ, re-

spectively). That is, δ and τ ′ are in the same (M11)σ′στ -orbit on V (Σ) \ σ′, σ, τ.

So there exists h ∈ (M11)σ′στ such that δh = τ ′. This implies that gh reverses

(σ′, σ, τ, τ ′) and hence ∆ is self-paired (Lemma 5.2.2). So there are exactly two

self-paired (M11)-orbits on proper 3-arcs of Σ, namely ∆1 := (σ′, σ, τ, τ ′)M11 for

τ ′ ∈ V (Σ) \ X(σ′, σ, τ), and ∆2 := (σ′, σ, τ, τ ′)M11 for τ ′ ∈ X(σ′, σ, τ) \ σ′, σ, τ.

Thus we get two 3-arc graphs, namely Ξi(M11) := Ξ(Σ,∆i) for i = 1, 2. Note that

|V (Σ) \ X(σ′, σ, τ)| = 3 and |X(σ′, σ, τ) \ σ′, σ, τ| = 6. So by Theorem 5.2.1(a)

each vertex of B(σ) other than στ is adjacent to three vertices of B(τ) in Ξ1(M11),

and adjacent to six vertices of B(τ) in Ξ2(M11). One can see that αα′, ββ ′ are ad-

jacent in Ξ1(M11) (Ξ2(M11), respectively) if and only if α′, α, β, β ′ are distinct and

β ′ ∈ V (Σ) \X(α′, α, β) (β ′ ∈ X(α′, α, β) \ α′, α, β, respectively). Thus, Ξ1(M11)

and Ξ2(M11) are the graphs defined in Proposition 5.1(e)(1) and (2) of [45], respec-

tively.

Example 6.6.2 The Mathieu group M22 of degree v+ 1 = 22 is the automorphism

group of the 3-(22, 6, 1) Steiner system D. We assume that the point set of D

78 Classification

is the same as the vertex set of Σ := K22. As in Example 6.6.1 above, we get

two 3-arc graphs of Σ, namely the graph Ξ1(M22) in which αα′, ββ ′ are adjacent

if and only if α′, α, β, β ′ are distinct and β ′ ∈ V (Σ) \ X(α′, α, β), and the graph

Ξ2(M22) in which αα′, ββ ′ are adjacent if and only if α′, α, β, β ′ are distinct and β ′ ∈

X(α′, α, β) \ α′, α, β, where X(α′, α, β) denotes the unique block of D containing

α′, α, β. These two graphs are the graphs defined in Proposition 5.1(d)(1) and (2)

of [45], respectively. Based on the same reason as in Example 6.6.1 one can see that

each vertex of B(α) other than αβ is adjacent to sixteen vertices of B(β) in Ξ1(M22),

and adjacent to three vertices of B(β) in Ξ2(M22).

Applying Theorem 5.2.3, the discussion in this chapter gives rise to the following

classification of all G-symmetric graphs Γ such that v = k+1 ≥ 3, D(B) contains no

repeated blocks and ΓB is a complete graph. This classification was obtained in [45]

by using a different approach. (By Theorem 4.3.2(b), in our case above GB must be

doubly transitive on B. So such graphs Γ are precisely those graphs studied in [45]

with the additional properties that val(ΓB) = v and v = k+1 ≥ 3. The objective of

[45] is to classify G-symmetric graphs with complete quotients such that the induced

action of G on each block of the G-invariant partition is doubly transitive.)

Theorem 6.6.1 Suppose that Γ is a G-symmetric graph which admits a nontrivial

G-invariant partition B of block size v = k + 1 ≥ 3 such that D(B) contains no

repeated blocks and ΓB is a complete graph, where G ≤ Aut(Γ). Then ΓB∼= Kv+1,

G is 3-transitive and faithful on B, and either Γ ∼= (v + 1) ·Kv with G an arbitrary

3-transitive permutation group of degree v+ 1, or one of the following (a)-(f) holds.

(a) Γ = (K(v + 1, 2))[K2], and G is either Sv+1 (v ≥ 3), or Av+1 (v ≥ 5), or

Mv+1 (v = 10, 11, 22, 23).

(b) (Γ, G) = (CR(v; x, n),PGL(2, v).〈ψt〉), where v = pe with p a prime and

e ≥ 1, x ∈ GF(v) \ 0, 1, n is a divisor of n(x), and t is a divisor of e with

gcd(n(x), t) = n.

(c) (Γ, G) = (TCR(v; x, n),M(t/2, v)), where v = pe with p an odd prime and

e ≥ 2 an even integer, x ∈ GF(v) \ 0, 1 with n(x) even and x − 1 a square of

GF(v), n is an even divisor of n(x), and t is a divisor of e with gcd(n(x), t) = n.

(d) Γ = Ξ1(d, 2) or Ξ2(d, 2) (defined in Example 6.5.1), v = 2d−1, where d ≥ 2,

and either G = AGL(d, 2) or d = 4 and G = Z42.A7.

Classification Theorem 79

(e) Γ = Ξ1(M11) or Ξ2(M11) (defined in Example 6.6.1), G = M11, and v = 11.

(f) Γ = Ξ1(M22) or Ξ2(M22) (defined in Example 6.6.2), G = M22, and v = 21.

In possibility (b) above, if v = 3 then PGL(2, 3) ∼= S4 and we get only one

cross-ratio graph CR(3; 2, 1) ∼= 3 · C4; if v = 4, then PGL(2, 4) ∼= A5 and we also

have a unique cross-ratio graph CR(4; t, 2) ∼= CR(4; t2, 2), which is isomorphic to the

dodecahedron (see [43, Example 2.4(a)]), where we set GF(4) = 0, 1, t, t2 = 1 + t.

80 Classification

Chapter 7

Almost covers of two-arctransitive graphs

Study it extensively, inquire into it accurately, think over it

carefully, sift it clearly, and practice it earnestly.

Confucius (551-479 B.C.), The Doctrine of the Mean 20

In this chapter we continue our study of the case where k = v− 1 ≥ 2 and D(B)

contains no repeated blocks. We notice that the possibilities for Γ[B,C] depend on

the pair (ΓB, G), and vice versa. For example, we have proved in Theorem 5.3.1

that the extreme case Γ[B,C] ∼= Kv−1,v−1 occurs if and only if ΓB is (G, 3)-arc

transitive. In this chapter we investigate the other extreme case for Γ[B,C], namely

Γ[B,C] ∼= (v−1) ·K2. In this case Γ is said to be an almost cover of ΓB (see Section

3.2). By using Theorem 6.6.1, we first classify all such graphs Γ in the case where in

addition ΓB∼= Kv+1 (Theorem 7.2.1). In the general case where ΓB 6∼= Kv+1 and ΓB

is connected, we find a surprising connection (Theorem 7.3.1) between such graphs

Γ and an interesting class of graphs, namely near-polygonal graphs. For an integer

n ≥ 4, a near n-gonal graph [75] is a pair (Σ, E) consisting of a connected graph Σ

of girth at least 4, together with a set E of n-cycles of Σ, such that each 2-arc of Σ is

contained in a unique member of E . In this case we also say that Σ is a near n-gonal

graph with respect to E . The main results in this chapter may be summarized as

follows.

Theorem 7.0.2 Suppose Γ is a G-symmetric graph admitting a nontrivial G-invari-

ant partition B of block size v ≥ 3 such that D(B) contains no repeated blocks, and

82 Almost covers

ΓB is connected and is almost covered by Γ, where G ≤ Aut(Γ). Then the following

(a)-(b) hold.

(a) If ΓB∼= Kv+1, then all possibilities for Γ and G are known explicitly.

(b) If ΓB 6∼= Kv+1, then for some even integer n ≥ 4, ΓB is a (G, 2)-arc transitive

near n-gonal graph with respect to a certain G-orbit on n-cycles of ΓB. Moreover,

any (G, 2)-arc transitive near n-gonal graph (where n ≥ 4 is even) with respect to a

G-orbit on n-cycles can appear as such a quotient ΓB.

In Section 7.4, we will study the special case where Γ is a G-locally primitive

almost cover of ΓB, and in the last section we will give criteria for testing when a

(G, 2)-arc transitive graph is near-polygonal. We will present and prove our results

in this chapter in terms of 3-arc graphs. By Theorem 5.2.3, the graphs Γ in Theorem

7.0.2 are precisely 3-arc graphs Ξ := Ξ(Σ,∆) which almost cover ΞB(Σ), where Σ is

a (G, 2)-arc transitive graph and ∆ is a self-paired G-orbit on Arc3(Σ). In this case

we also say that Ξ almost covers Σ since ΞB(Σ)∼= Σ (Theorem 5.2.1(b)).

7.1 Preliminaries

As in the last Chapter, we will denote an arc (σ, τ) of a graph Σ by στ when this

is convenient and unlikely to cause confusion. The following simple lemma follows

from Theorem 5.2.1(a).

Lemma 7.1.1 Let Σ be a connected (G, 2)-arc transitive graph. Let ∆ be a self-

paired G-orbit on Arc3(Σ), and let (τ, σ, σ′, τ ′) ∈ ∆. Then Ξ(Σ,∆) almost covers Σ

if and only if τ ′ is fixed by Gτσσ′ (that is, Gτσσ′ = Gτσσ′τ ′).

Let Γ = Ξ(Σ,∆) be a 3-arc graph of the (G, 2)-arc transitive graph Σ. If Γ

almost covers Σ, then for each τ ∈ Σ(σ) \ σ′ there exists a unique τ ′ ∈ Σ(σ′) \ σ

such that (τ, σ, σ′, τ ′) ∈ ∆, and hence τ 7→ τ ′ defines a bijection from Σ(σ) \ σ′ to

Σ(σ′) \ σ. Note that this bijection depends on ∆. Since there will be no danger

of confusion, we will denote it just by φσσ′ . Recall that a G-vertex-transitive graph

Σ is (G, 2)-arc transitive if and only if Gσ is 2-transitive on Σ(σ) for σ ∈ V (Σ) (see

Lemma 3.1.1(b)).

Preliminaries 83

Lemma 7.1.2 Let Σ be a connected (G, 2)-arc transitive graph, where G ≤ Aut(Σ).

Let ∆ be a self-paired G-orbit on Arc3(Σ) and let στ be an arc of Σ. Suppose that

the 3-arc graph Γ := Ξ(Σ,∆) almost covers Σ. Then the following (a)-(d) hold:

(a) The actions of Gσ on B(σ) and Σ(σ) are permutationally equivalent, doubly

transitive and faithful.

(b) The actions of Gστ on Σ(σ) \ τ and on Γ(στ) are permutationally equiv-

alent, where Γ(στ) is the neighbourhood of στ in Γ. In particular, Γ is G-locally

primitive if and only if Gσ is 2-primitive on Σ(σ); and Gστ is regular on Γ(στ) if

and only if Gσ is sharply 2-transitive on Σ(σ).

(c) φ−1στ = φτσ.

(d) (φστ (ε))g = φσgτg(εg) for ε ∈ Σ(σ)\τ and g ∈ G. In particular, the actions

of Gστ on Σ(σ) \ τ and Σ(τ) \ σ are permutationally equivalent with respect to

φστ .

Proof (a) By Lemma 5.2.1(d), the actions of Gσ on B(σ) and Σ(σ) are permuta-

tionally equivalent with respect to the bijection B(σ) → Σ(σ) defined by στ 7→ τ for

τ ∈ Σ(σ). Since Σ is (G, 2)-arc transitive, these actions are doubly transitive. The

faithfulness follows from Lemma 5.2.1(a), Theorem 4.3.1(d) and Lemma 4.1.2(a).

(b) For each ε ∈ Σ(σ)\τ, let λ(ε) denote the unique vertex in B(ε) adjacent to

στ in Γ. (The existence of λ(ε) follows from Theorem 5.2.1(a).) Then λ establishes

a bijection from Σ(σ) \ τ to Γ(στ). Clearly, (λ(ε))g ∈ Γ(στ) for g ∈ Gστ . Since

λ(ε) ∈ B(ε), we have (λ(ε))g ∈ (B(ε))g = B(εg) and hence λ(εg) = (λ(ε))g by

the definition of λ. Thus, the actions of Gστ on Σ(σ) \ τ and on Γ(στ) are

permutationally equivalent with respect to λ. From this the last two assertions in

(b) follow immediately.

(c) This is obvious from the definition of φστ .

(d) For (ε, σ, τ, η) ∈ ∆ and g ∈ G, since ∆ is G-invariant we have (εg, σg, τ g, ηg) ∈

∆ and so (φστ (ε))g = ηg = φσgτg(εg) (by the definitions of φστ and φσgτg). In

particular, (φστ (ε))g = φστ (ε

g) for g ∈ Gστ and hence the assertion in the last

sentence of (d) is true. 2

The following result was advertised in Section 5.3. It shows that, if Gσ is sharply

2-transitive on Σ(σ) (that is, Gσ holds the “weakest” 2-transitivity on Σ(σ)), then

all the 3-arc graphs of Σ are forced to be almost covers of Σ.

84 Almost covers

Proposition 7.1.1 Suppose that Σ a (G, 2)-arc transitive graph of valency v ≥ 3

such that Gσ is sharply 2-transitive on Σ(σ) for σ ∈ V (Σ). Then, for every self-

paired G-orbit ∆ on Arc3(Σ), the 3-arc graph Γ := Ξ(Σ,∆) is an almost cover of Σ

and Gστ is regular on the neighbourhood Γ(στ) of στ ∈ V (Γ) in Γ.

Proof Let στ be an arc of Σ. Then the sharp 2-transitivity of Gσ on Σ(σ) implies

that Gστ is regular on Σ(σ) \ τ, and hence we have |Gστ | = |Σ(σ)| − 1. Since

Γ(στ) contains exactly s points of each block B(δ) for δ ∈ Σ(σ) \ τ, where s is

the valency of the bipartite graph Γ[B(σ), B(δ)] as defined in Section 3.2, we then

have |Γ(στ)| = s(|Σ(σ)| − 1) = s|Gστ |. On the other hand, since Gστ is transitive

on Γ(στ), by the orbit-stabilizer property (see Lemma 2.1.1(c)), |Γ(στ)| is a divisor

of |Gστ |. So we have s = 1, that is, Γ[B(σ), B(τ)] = (v−1) ·K2, and hence Γ almost

covers Σ. Since Gστ is regular on Σ(σ) \ τ, from Lemma 7.1.2(b) we know that

Gστ is also regular on Γ(στ). 2

For a near n-gonal graph (Σ, E), the cycles in E are called basic cycles of (Σ, E).

We use C(σ, τ, ε) to denote the unique basic cycle of (Σ, E) containing a given 2-arc

(σ, τ, ε) of Σ. We also use Arc3(Σ, E) to denote the set of all 3-arcs of Σ which are

contained in some basic cycle of (Σ, E). Since the number of 2-arcs contained in a

basic cycle of (Σ, E) is 2n and since each 2-arc is contained in a unique basic cycle,

we have 2n|E| = |Arc2(Σ)| = v(v − 1)|V (Σ)| = (v − 1)|Arc(Σ)|, where v = val(Σ).

So n and |E| are connected by

|E| = (v − 1)|Arc(Σ)|/2n.

Any subgroup G ≤ Aut(Σ) induces an action on n-cycles of Σ, and if E is G-

invariant, then G induces an action on E . A circulant is a Cayley graph Cay(Zn, S)

with vertex set the additive group Zn of integers modulo n in which x, y ∈ Zn are

adjacent if and only if x − y ∈ S, where S is a subset of Zn such that 0 6∈ S and

−S := −x : x ∈ S is equal to S.

Lemma 7.1.3 Suppose (Σ, E) is a finite (G, 2)-arc transitive near n-gonal graph.

Then the following statements (a)-(c) are equivalent:

(a) E is G-invariant.

(b) E is a G-orbit on n-cycles of Σ.

Preliminaries 85

(c) Arc3(Σ, E) is a self-paired G-orbit on Arc3(Σ).

Moreover, if one of these occurs, then the following (d)-(e) hold:

(d) Any element of G fixing a 2-arc (σ, τ, ε) of Σ must fix each vertex in C(σ, τ, ε).

(e) The subgraph of Σ induced by the vertex set of a basic cycle of (Σ, E) is

isomorphic to a circulant graph Cay(Zn, S), for some S with 1 ∈ S. Moreover, each

such basic cycle is chordless (that is, Cay(Zn, S) ∼= Cn) unless, for adjacent vertices

σ, τ of Σ, either

(i) Gτ is sharply 2-transitive on Σ(τ) (and hence |Σ(τ)| is a prime power); or

(ii) Gστ is imprimitive on Σ(τ) \ σ.

Proof The equivalence of (a) and (b) is obvious since each 2-arc of Σ lies in a

unique cycle of E . If (a) holds, then Arc3(Σ, E) is a G-orbit on Arc3(Σ). Moreover,

in this case Arc3(Σ, E) is also self-paired. In fact, for (σ, τ, ε, η) ∈ Arc3(Σ, E) there

exists g ∈ G such that (σ, τ, ε)g = (η, ε, τ) as Σ is (G, 2)-arc transitive. Thus,

(C(σ, τ, ε))g = C(η, ε, τ). But C(σ, τ, ε) is the unique basic cycle containing (σ, τ, ε),

and it is also the unique basic cycle containing (η, ε, τ). So g fixes C(σ, τ, ε) and

ηg = σ, implying (η, ε, τ, σ) = (σ, τ, ε, η)g ∈ Arc3(Σ, E). Hence Arc3(Σ, E) is self-

paired. Thus (a) implies (c). Conversely suppose that (c) holds. Let

C(σ0, σ1, σ2) = (σ0, σ1, σ2, . . . , σn−1, σ0)

be the basic cycle of (Σ, E) containing the 2-arc (σ0, σ1, σ2), and let g ∈ G. For

each i = 0, 1, . . . , n− 1 (subscripts modulo n here and in the remaining part of the

proof), it follows from (c) that both (σgi−1, σgi , σ

gi+1, σ

gi+2) and (σgi , σ

gi+1, σ

gi+2, σ

gi+3)

lie in basic cycles, and they must lie in the same basic cycle since these two 3-arcs

have the 2-arc (σgi , σgi+1, σ

gi+2) in common and since each 2-arc of Σ is contained in a

unique basic cycle of (Σ, E). Since this is true for all i, it follows that (C(σ0, σ1, σ2))g

must be a basic cycle of (Σ, E) and hence (c) implies (a).

In the remainder of this proof, we suppose E is G-invariant, so both (b) and

(c) hold. Thus the vertex sets of the basic cycles of (Σ, E) induce mutually isomor-

phic subgraphs. If g ∈ G fixes the 2-arc (σ0, σ1, σ2), then it fixes the basic cycle

C(σ0, σ1, σ2) and, since g fixes each of σ1, σ2, it follows that g must fix σ3. Induc-

tively, one can see that g fixes each vertex in C(σ0, σ1, σ2) and thus (d) is proved.

In proving (e), we set V := σ0, σ1, σ2, . . . , σn−1, the vertex set of C(σ0, σ1, σ2),

86 Almost covers

and denote by Σ1 the subgraph of Σ induced by V . Since Σ is (G, 2)-arc transitive,

there exists h ∈ G such that (σn−1, σ0, σ1)h = (σ0, σ1, σ2). Since E is G-invariant it

follows that h fixes V setwise and leaves C(σ0, σ1, σ2) invariant. The only element of

Aut(Σ1) which leaves C(σ0, σ1, σ2) invariant and maps (σn−1, σ0, σ1) to (σ0, σ1, σ2)

is the rotation ρ : σi 7→ σi+1, for all i. Thus the permutation hV of V induced by

h is ρ, and by [6, Lemma 16.3], since 〈ρ〉 ∼= Zn is regular on V , Σ1 is isomorphic to

a circulant Cay(Zn, S) for some S. Since σi is adjacent to σi+1, we have 1 ∈ S and

the first part of (e) is proved. In proving the second part of (e), we assume that

C(σ0, σ1, σ2) contains a chord. Since the group induced on C(σ0, σ1, σ2) contains ρ,

it follows that σ1 is adjacent to some vertex σi with i 6= 0, 2, that is to say, σ1, σi

is a chord; and the set X := fixΣ(σ1)\σ0(Gσ0σ1σ2) contains both σ2 and σi. On

the other hand, the (G, 2)-arc transitivity of Σ implies that Gσ0σ1 is transitive on

Σ(σ1) \ σ0, and the stabilizer Gσ0σ1σ2 (which fixes C(σ0, σ1, σ2) pointwise) fixes

|X| ≥ 2 points of Σ(σ1) \ σ0. By Lemma 2.2.1, X is a block of imprimitivity for

Gσ0σ1 in Σ(σ1) \ σ0. Hence either X = Σ(σ1) \ σ0 or X induces a nontrivial

Gσ0σ1-invariant partition of Σ(σ1) \ σ0. In the former case the possibility (i) in

(e) occurs; whilst in the latter case the possibility (ii) in (e) occurs. Note that if (i)

occurs then by [95, pp. 23] |Σ(σ1)| must be a prime power. 2

7.2 Almost covers of complete graphs

In this section we assume that Σ := Kv+1 is a (G, 2)-arc transitive graph of valency

v ≥ 3, where G ≤ Aut(Σ). So, by Lemma 6.1.1, G is 3-transitive on V (Σ) and thus

is one of the groups listed at the beginning of the previous chapter. In the following

we will show that almost covers Ξ(Σ,∆) of Σ exist (where ∆ is a self-paired G-orbit

on Arc3(Σ)), and the goal of this section is to determine all of them. Recall that in

Theorem 6.6.1 we have classified all 3-arc graphs of Σ. So what we need to do here

is to determine which of them are almost covers of Σ.

By Lemmas 5.2.2 and 7.1.1, aG-orbit ∆ := (τ, σ, σ′, τ ′)G on Arc3(Σ) is self-paired

and Ξ(Σ,∆) almost covers Σ if and only if στ, σ′τ ′ can be reversed by an element ofG

and Gτσσ′ = Gτσσ′τ ′ . In the case where (τ, σ, σ′, τ ′) is a 3-cycle of Σ, we get a unique

graph Ξ(Σ,∆) ∼= (v+1) ·Kv which almost covers Σ (see Example 6.1.1). Therefore,

we may assume in the following that (τ, σ, σ′, τ ′) is proper. Then the requirement

Almost Covers of Complete Graphs 87

Gτσσ′ = Gτσσ′τ ′ implies that either v+1 = |V (Σ)| = 4, or G is 3- but not 4-transitive

on V (Σ). Hence the groups listed in Example 6.1.2 and the corresponding graph

(K(v + 1, 2))[K2] therein can be excluded. From the discussion in Examples 6.6.1

and 6.6.2, we can also exclude the Mathieu group M11 of degree 12 and the Mathieu

group M22 of degree 22. Therefore, from the list at the beginning of Chapter 6,

only AGL(d, 2) (v = 2d − 1 ≥ 3), Z42.A7 (v = 15) and the 3-transitive subgroups of

PΓL(2, v) (v is a prime power) can satisfy the conditions in Lemmas 5.2.2 and 7.1.1

for a proper 3-arc (τ, σ, σ′, τ ′) of Σ. The 3-transitive subgroups of PΓL(2, v) were

described in Theorem 6.2.1, and by Corollary 6.4.1(a) the 3-arc graphs arising from

such groups are (twisted or untwisted) cross-ratio graphs. The following example

tells us when a cross-ratio graph is an almost cover of Σ.

Example 7.2.1 Cross-ratio graphs which almost cover Kv+1. Let v = pe ≥ 3 be a

prime power. For Γ = CR(v; x, n) and G = PGL(2, v).〈ψn〉 or for Γ = TCR(v; x, n)

and G = M(n/2, v) (for proper x and n), the vertex set of Γ admits the G-invariant

partition B := B(v) (see Section 6.3 for definition) such that the block size of

D(B) is equal to v − 1, where B ∈ B. By Corollary 6.3.1, for distinct blocks

B,C of B, the bipartite subgraph Γ[B,C] has valency n(x)/n. So Γ almost covers

Σ if and only if n(x) = n. Thus, by Corollary 6.4.1(a), the only 3-arc graphs

Ξ(Σ,∆) of Σ = Kv+1 which almost cover Σ are CR(v; x, n(x)) and TCR(v; x, n(x)),

for x ∈ GF(v) \ 0, 1, where ∆ is a self-paired G-orbit on Arc3(Σ). Note that

gcd(n(x), t) = n(x) implies that n(x) is a divisor of t. So by Corollary 6.4.1(b), the

only 3-transitive subgroups H of PΓL(2, v) such that CR(v; x, n(x)) is H-symmetric

have the form H = PGL(2, v).〈ψt〉, where t is a divisor of e and a multiple of n(x).

Similarly, the only 3-transitive subgroups H of PΓL(2, v) such that TCR(v; x, n(x))

is H-symmetric have the form H = M(t/2, v), where t is an even divisor of e and a

multiple of n(x).

The next example determines the 3-arc graphs arising from AGL(d, 2) and Z42.A7

which almost cover Σ.

Example 7.2.2 In Example 6.5.1 we have shown that, apart from the graph (v +

1) ·Kv, there are only two 3-arc graphs of Σ arising from the group G = AGL(d, 2)

(v = 2d − 1 ≥ 3) or the group G = Z42.A7 (if d = 4), namely Ξi(d, 2) for i = 1, 2. It

follows from the definition that Ξ1(d, 2) is, and Ξ2(d, 2) is not, an almost cover of Σ.

88 Almost covers

The analysis above leads to the following classification theorem.

Theorem 7.2.1 Suppose Σ = Kv+1 is a (G, 2)-arc transitive complete graph, where

v ≥ 3 and G ≤ Aut(Σ). Suppose further that Γ = Ξ(Σ,∆) almost covers Σ, where ∆

is a self-paired G-orbit on Arc3(Σ). Then either Γ ∼= (v+1) ·Kv with G an arbitrary

3-transitive permutation group of degree v + 1, or (Γ, G) is one of the following

(where in (a), (b), v = pe with p a prime and e ≥ 1):

(a) (CR(v; x, n(x)),PGL(2, v).〈ψt〉), where x ∈ GF(v) \ 0, 1, and t is a divisor

of e and a multiple of n(x);

(b) (TCR(v; x, n(x)),M(t/2, v)), where p is odd, e is even, x ∈ GF(v) \ 0, 1

with n(x) even and x − 1 a square of GF(v), and t is an even divisor of e and a

multiple of n(x);

(c) (Ξ1(d, 2),AGL(d, 2)), where v = 2d − 1 ≥ 3; or

(d) (Ξ1(4, 2), Z42.A7), where v = 15.

7.3 Almost covers of non-complete graphs

Now we discuss the general case where Σ is a connected, non-complete, (G, 2)-arc

transitive graph with valency v ≥ 3. Then girth(Σ) ≥ 4 by Lemma 6.1.1. The main

result in this case is the following theorem which, together with Theorems 5.2.3 and

7.2.1, yields a proof of Theorem 7.0.2 stated at the beginning of this chapter.

Theorem 7.3.1 Suppose that Σ is a connected (G, 2)-arc transitive graph with va-

lency v ≥ 3 and that Σ 6∼= Kv+1. Then Σ is almost covered by a 3-arc graph Ξ(Σ,∆)

of Σ if and only if, for some even integer n ≥ 4, Σ is a near n-gonal graph with

respect to a G-orbit E of n-cycles of Σ, and in this case we have ∆ = Arc3(Σ, E),

the set of all 3-arcs of Σ contained in the n-cycles in E .

Proof Suppose Σ is almost covered by a 3-arc graph Γ := Ξ(Σ,∆) of Σ, where

∆ is a self-paired G-orbit on Arc3(Σ). Recall that, for adjacent vertices σ, σ′ of

Σ, we use φσσ′ to denote the the bijection from Σ(σ) \ σ′ to Σ(σ′) \ σ such

that φσσ′(τ) = τ ′ precisely when (τ, σ, σ′, τ ′) ∈ ∆. Let (σ0, σ1, σ2) be a 2-arc of

Σ. Set σ3 := φσ1σ2(σ0), and inductively define σi+2 := φσiσi+1(σi−1) for i ≥ 1.

Then we get a sequence σ0, σ1, σ2, . . . , σi−1, σi, σi+1, σi+2, . . . of vertices of Σ such

Almost Covers of Non-complete Graphs 89

that (σi−1, σi, σi+1, σi+2) ∈ ∆ for each i ≥ 1. Our assumption Σ 6∼= Kv+1 implies that

girth(Σ) ≥ 4 (Lemma 6.1.1) and hence all such 3-arcs (σi−1, σi, σi+1, σi+2) are proper,

that is, any four consecutive vertices in this sequence are pairwise distinct. Since

Σ has a finite number of vertices, the sequence must eventually contain repeated

vertices. Let σn be the first vertex in the sequence that coincides with one of the

preceding vertices. We claim that σn must coincide with σ0. Suppose to the contrary

that σn = σℓ for some ℓ such that 1 ≤ ℓ < n. Then since Σ is (G, 2)-arc transitive,

there exists g ∈ G such that (σℓ, σℓ+1, σℓ+2)g = (σ0, σ1, σ2). From Lemma 7.1.2(d),

we have σgℓ+3 = φσgℓ+1

σgℓ+2

(σgℓ ) = φσ1σ2(σ0) = σ3. Inductively we have that σgℓ+i = σi

for each i ≥ 0. In particular, σgn = σgℓ+(n−ℓ) = σn−ℓ. But since σn = σℓ, we have

σn−ℓ = σgn = σgℓ = σ0, contradicting the minimality of n. Therefore we must have

σn = σ0. Thus, each 2-arc (σ0, σ1, σ2) of Σ determines a unique (undirected) n-

cycle C(σ0, σ1, σ2) := (σ0, σ1, σ2, . . . , σn−1, σ0) of Σ. Note again that n ≥ 4 since

girth(Σ) ≥ 4.

Set τ := φσ1σ0(σ2), then we have σ2 = φσ0σ1(τ) by Lemma 7.1.2(c). We claim that

τ must coincide with σn−1. For the 2-arc (τ, σ0, σ1), the construction in the previous

paragraph will give the sequence τ, σ0, σ1, σ2, . . . , σn−1, σn = σ0, and since the first

repeated vertex is the same as the starting vertex τ , it follows that τ = σn−1. Sim-

ilarly, one can show that σn−2 = φσ0σn−1(σ1) and hence σ1 = φσn−1σ0(σn−2). There-

fore, reading the subscripts modulo n (here and in the remainder of this section),

we have σi+2 = φσiσi+1(σi−1) and hence σi−1 = φσi+1σi

(σi+2) for each i ≥ 1 (Lemma

7.1.2(c)). This implies that the 2-arcs (σi−1, σi, σi+1) and (σi+1, σi, σi−1) contained in

C(σ0, σ1, σ2) (for i ≥ 1) also determine the same n-cycle C(σ0, σ1, σ2). By definition

of C(σ0, σ1, σ2) and by Lemma 7.1.2(d), we have C(σg0 , σg1, σ

g2) = (C(σ0, σ1, σ2))

g for

g ∈ G and hence E := C(σ, τ, ε) : (σ, τ, ε) ∈ Arc2(Σ) is G-invariant and each

2-arc lies in a unique cycle of E . By the (G, 2)-arc transitivity of Σ, the length n of

C(σ, τ, ε) is independent of the choice of (σ, τ, ε) and G is transitive on E . Thus E is

a G-orbit on n-cycles of Σ and Σ is a near n-gonal graph with respect to E . More-

over, the argument above shows that ∆ = Arc3(Σ, E). In particular, in the sequence

σ0σ1, σ1σ0, σ2σ3, σ3σ2, . . . , σ2i−2σ2i−1, σ2i−1σ2i−2, σ2iσ2i+1, σ2i+1σ2i, . . . of vertices of Γ,

for each i, the (2i − 1)-st vertex σ2i−2σ2i−1 and the 2i-th vertex σ2i−1σ2i−2 are not

adjacent, while the 2i-th vertex and the (2i+ 1)-st vertex σ2iσ2i+1 are adjacent. By

the definition of n, the n-th vertex of this sequence is σn−1σn−2, and it is adjacent

90 Almost covers

to σ0σ1 (= σnσn+1) since (σi−1, σi, σi+1, σi+2) ∈ ∆ for each i (subscripts modulo n).

It follows that n must be an even integer.

To prove the “if” part of the theorem, suppose that (Σ, E) is a (G, 2)-arc transitive

near n-gonal graph with valency v ≥ 3 and E is a G-orbit on n-cycles of Σ, for some

even n ≥ 4. Then by Lemma 7.1.3, ∆ := Arc3(Σ, E) is a self-paired G-orbit on

Arc3(Σ). Let Γ := Ξ(Σ,∆) and let (τ, σ, σ′, τ ′) ∈ ∆. Then στ ∈ B(σ) is adjacent

to σ′τ ′ ∈ B(σ′) in Γ. If στ is adjacent in Γ to a second vertex, say σ′ε′, of B(σ′),

then (τ, σ, σ′, τ ′), (τ, σ, σ′, ε′) are distinct 3-arcs in ∆ and hence the 2-arc (τ, σ, σ′)

is contained in two distinct basic cycles of (Σ, E). This contradiction shows that

Γ[B(σ), B(σ′)] ∼= (v − 1) ·K2 and hence Γ almost covers ΓB(Σ). 2

Remark 7.3.1 By Lemma 7.1.3(e), the vertex set of each basic cycle of (Σ, E) in

Theorem 7.3.1 induces a circulant subgraph of Σ, and these basic cycles are chordless

unless either (e)(i) or (e)(ii) in that lemma occurs. This latter fact is interesting from

a combinatorial point of view. The following example shows that the basic cycles of

(Σ, E) may contain chords. It also provides an example of such a graph Σ with the

smallest valency (namely 3) and shows that the near n-gonal graph (Σ, E) occurring

in Theorem 7.3.1 is not necessarily an n-gonal graph. (A near n-gonal graph is said

to be an n-gonal graph [75] if n is equal to the girth of the graph.) Moreover, it

shows that the graph Ξ(Σ,∆) may not be connected, even if Σ is connected and

(G, 2)-arc transitive.

Example 7.3.1 Let Σ be the complete bipartite graph K3,3 with vertex set 0, 1, 2,

3, 4, 5 and bipartition (0, 2, 4, 1, 3, 5). We will show that there exists a unique

subgroup G ≤ Aut(Σ) such that Σ is a (G, 2)-arc transitive near 6-gonal graph with

respect to a G-orbit E of 6-cycles of Σ. By the definition of near polygonal graphs,

one can easily check that

E1 := (0, 1, 2, 3, 4, 5, 0), (0, 5, 2, 1, 4, 3, 0), (0, 1, 4, 5, 2, 3, 0)

and

E2 := (0, 1, 2, 5, 4, 3, 0), (0, 3, 2, 1, 4, 5, 0), (0, 1, 4, 3, 2, 5, 0)

are the only possible sets E of 6-cycles of Σ such that (Σ, E) is a near 6-gonal graph.

On the other hand, we have Aut(Σ) = S3 wrS2∼= 〈(024), (02), (01)(23)(45)〉 and

Almost Covers of Non-complete Graphs 91

again it is easily checked that (024) and (01)(23)(45) fix E1 and E2 setwise, whilst

(02) interchanges E1 and E2. Thus Aut(Σ) interchanges E1 and E2 and so a subgroup

G of Aut(Σ) with index 2 fixes E1 and E2 setwise. We have seen that G contains

H = 〈(024), (01)(23)(45)〉 ∼= A3 wrS2, but does not contain (02). Thus |G : H| = 2.

The element (13) is the conjugate of (02) by (01)(23)(45), and hence (13) ∈ Aut(Σ)

and (13) interchanges E1 and E2. Therefore (02)(13) fixes E1 and E2 setwise and does

not lie in H , so G = 〈H, (02)(13)〉. It is easy to check that G is transitive on the

2-arcs of Σ, and hence (Σ, Ei) is a (G, 2)-arc transitive near 6-gonal graph for i = 1

and i = 2. If Σ is (K, 2)-arc transitive and K preserves the Ei, then K ≤ G and

|K| is divisible by the number of 2-arcs, that is, by 36. Hence K = G. Finally, for

∆i := Arc3(Σ, Ei), i = 1, 2, we have Ξ(Σ,∆i) ∼= 3 · C6 (see Figure 4 below).

Figure 4 Γ = 3 · C6,Σ = K3,3

The following proposition shows further that the graph Σ in Example 7.3.1 is

the only connected trivalent non-complete graph which is (G, 2)-arc transitive and

near n-gonal for an even integer n such that the basic cycles have chords.

Proposition 7.3.1 Suppose Σ is a connected, (G, 2)-arc transitive, trivalent graph

and Σ 6∼= K4. Suppose ∆ is a self-paired G-orbit on Arc3(Σ) such that Γ := Ξ(Σ,∆)

almost covers Σ. Then Σ is a near n-gonal graph with respect to some G-orbit E

of n-cycles (and n is even). Moreover the cycles in E have chords if and only if

Σ ∼= K3,3, Γ ∼= 3 ·C6, and E ∼= E1 or E2, where G, E1 and E2 are as in Example 7.3.1.

92 Almost covers

Proof By Theorem 7.3.1, Σ is a near n-gonal graph with respect to some G-orbit

E of n-cycles for an even integer n ≥ 4. So we need only to prove that the cycles

in E have chords if and only if Σ,Γ, G, E are as claimed. The “if” part was in fact

proved in Example 7.3.1. We prove the “only if” part in the following.

Suppose σ0, σℓ is a chord of the basic cycle C(σ0, σ1, σ2) := (σ0, σ1, σ2, . . . , σn−1,

σ0). Then σi, σi+ℓ is a chord of C(σ0, σ1, σ2) for each i (by Lemma 7.1.3(e)).

Since Σ is trivalent and connected, the only possibility is ℓ = n/2 and Σ ∼=

Cay(Zn, 1, ℓ, n − 1). Since Σ 6∼= K4, we have ℓ ≥ 3. Now the unique n-cycle

C(σℓ, σ0, σ1) containing (σℓ, σ0, σ1) must be the following sequence of vertices: σℓ, σ0,

σ1, σℓ+1, σℓ+2, σ2, σ3, σℓ+3, σℓ+4, . . .. If ℓ is even, this sequence does not even form an

n-cycle since it never returns to the vertex σℓ. (Once we arrive at σℓ−1, the next ver-

tex in the sequence is σn−1 and from σn−1 the sequence returns to σ0. For example,

if ℓ = 4, then the sequence is the 7-cycle (σ0, σ1, σ5, σ6, σ2, σ3, σ7, σ0).) So ℓ is odd,

and in this case the sequence does give an n-cycle. By the (G, 2)-arc transitivity of

Σ, there exists g ∈ G such that (σn−1, σ0, σ1)g = (σℓ, σ0, σ1). From Lemma 7.1.2(d),

we have (C(σn−1, σ0, σ1))g = C(σℓ, σ0, σ1). Therefore, σg0 = σ0, σ

g1 = σ1, σ

gn−1 =

σℓ, σgn−3 = σn−1. Since σ0, σℓ are adjacent, we know that σg0 and σgℓ are adjacent,

and hence the only possibility for σgℓ is σgℓ = σn−1 (note that σgℓ 6= σg1 = σ1, σgℓ 6=

σgn−1 = σℓ). But σgn−3 = σn−1 as mentioned above, so we get σgℓ = σgn−3 and hence

σℓ = σn−3. Therefore, n = 6 and hence Σ = Cay(Z6, 1, 3, 5) = K3,3. From the

discussion in Example 7.3.1, we then have Γ = 3 · C6, E is either E1 or E2, and G is

the group 〈(024), (02)(13), (01)(23)(45)〉. 2

7.4 Locally primitive almost covers

In this section, we examine an important special case which was the original moti-

vation for the study in this chapter. Recall that if Γ is a G-locally primitive graph

admitting a nontrivial G-invariant partition B of block size v = k + 1 ≥ 3 such

that ΓB is connected, then D(B) contains no repeated blocks (Corollary 3.3.1) and

ΓB is almost covered by Γ (Corollary 3.2.1). By Theorem 5.2.3, Γ = Arc∆(Σ) for

some self-paired G-orbit ∆ on 3-arcs of Σ := ΓB, and B is identical with B(Σ) (see

the comments before Theorem 5.2.2). Since Γ is G-locally primitive, from Corol-

lary 4.3.1, GB is 2-primitive on B and Σ(B) (this result also follows from Lemma

Locally Primitive Almost Covers 93

7.1.2(a) and (b)). If in addition girth(Σ) = 3 (that is, Σ ∼= Kv+1, see Lemma 6.1.1),

then G is 3-primitive on B and the argument in the proof of [43, Theorem 5.4]

from (Line, Page) = (25, 534) to (12, 535) shows that we get the possibilities for

(Γ, G) listed in part (a) and the second half of part (b) of [43, Theorem 5.4]. (It

also comes from the classification of 3-primitive groups and the discussion in Exam-

ple 7.2.1.) However, in the general case where girth(Σ) ≥ 4, the argument in [43,

lines 33-41, pp. 534] should be modified since the block D therein is not adjacent

to C. In this case, as mentioned in Theorem 7.0.2(b), Σ is a near n-gonal graph

with n ≥ 4 and n even. Moreover, GΣ(B)B is 2-primitive. Hence if basic cycles of Σ

have chords, then by Lemma 7.1.3(e), GB is sharply 2-primitive on Σ(B). Hence

GB is also sharply 2-primitive on B, and so, v is a prime power and, for α ∈ B,

GB\αα = Zv−1 with v − 1 a prime. Hence either v = 3, or v = 2p for a prime p

with q = 2p − 1 a Mersenne prime. In the former case Proposition 7.3.1 implies

that Σ = K3,3, Γ = 3 · C6, and G and E are as in Example 7.3.1. In the latter case

K := g ∈ GBB : g = 1 or g fixes no vertex in B is a regular normal elementary

abelian subgroup of GBB ([26, Theorem 3.4B, pp.88]) and so GB

B = (Z2)p.Zq. Theo-

rems 5.2.3 and 7.3.1 and the argument above imply the following corollary, which is

an amended form of [43, Theorem 5.4].

Corollary 7.4.1 Suppose that Γ is a G-locally primitive graph admitting a nontriv-

ial G-invariant partition B of block size v = k + 1 ≥ 3 such that ΓB is connected.

Then ΓB is a (G, 2)-arc transitive graph of valency v and is almost covered by Γ, the

actions of GB on B and ΓB(B) are permutationally equivalent and 2-primitive, and

the following (a)-(b) hold.

(a) If ΓB∼= Kv+1, then either

(i) Γ ∼= (v+1)·Kv and G is one of the following: Sv+1 (v ≥ 3), Av+1 (v ≥ 4),

Mv+1 (v = 10, 11, 22, 23), M11 (v = 11), PGL(2, 2p) (v = 2p with 2p − 1

a Mersenne prime), or

(ii) Γ ∼= CR(3; 2, 1) = 3 · C4 and G = PGL(2, 3) (v = 3), or

(iii) Γ ∼= CR(2p; x, n(x)) and G = PGL(2, 2p) (v = 2p) for some x ∈ GF(2p) \

0, 1 with 2p − 1 a Mersenne prime.

94 Almost covers

(b) If ΓB 6∼= Kv+1, then for some even integer n ≥ 4, ΓB is a near n-gonal graph

with respect to a certain G-orbit E on n-cycles of ΓB, and Γ ∼= Ξ(ΓB,∆) for

∆ := Arc3(ΓB, E). Moreover, each basic cycle of (ΓB, E) is chordless unless

GBB is sharply 2-primitive and either

(i) v = 3, ΓB∼= K3,3, Γ ∼= 3 · C6, and G and E are as in Example 7.3.1, or

(ii) GBB = (Z2)

p.Zq and v = 2p with p a prime and q = 2p − 1 a Mersenne

prime.

The smallest v in part (b)(ii) above is v = 22 = 4. In this case we have GBB =

(Z2)2.Z3 and a similar argument as in the proof of Proposition 7.3.1 shows that, if

the basic cycles of (ΓB, E) have chords, then the subgraph induced by the vertex set

of each basic cycle is isomorphic to the circulant Cay(Zn, S) for S = 1, n/2, n− 1.

7.5 Two-arc transitive near-polygonal graphs

Let us review briefly the group-theoretic method for constructing 2-arc transitive

graphs. Let G be a finite group. A subgroup H of G is said to be core-free if its

core in G (see Example 2.1.2 for the definition) is equal to the identity, that is,⋂

g∈GHg = 1. For such a subgroup H and for a 2-element g of G with g 6∈ NG(H)

(where NG(H) is the normalizer of H in G), define Γ(G,H,HgH) = (V ∗, E∗) to be

the graph such that

V ∗ := [G : H ] = Hx : x ∈ G, E∗ := Hx,Hy : xy−1 ∈ HgH.

Sabidussi [76] (and others, see e.g. [56]) proved that Γ(G,H,HgH) is a G-symmetric

graph, and that any G-symmetric graph is isomorphic to Γ(G,H,HgH) for a certain

core-free subgroup H and 2-element g of G. Moreover, the graph Γ(G,H,HgH) is

connected if and only if 〈H, g〉 = G. By refining this classic result, Fang and Praeger

[33, Theorem 2.1] (see also [70, Theorem 11.1]) gave the following construction of

(G, 2)-arc transitive graphs.

Theorem 7.5.1 ([33, Theorem 2.1]) Let G be a finite group with a core-free sub-

group H and a 2-element g. Then the graph Γ(G,H,HgH) is a finite, connected,

Near-polygonal Graphs 95

(G, 2)-arc transitive graph with G transitive on vertices (acting by right multiplica-

tion as defined in Example 2.1.2) if and only if

g 6∈ NG(H), g2 ∈ H, 〈H, g〉 = G,

and the action of H on [H : H ∩Hg] by right multiplication is doubly transitive.

Let F denote the class of G-symmetric graphs Γ such that k = v − 1 ≥ 2, ΓB is

connected and (G, 2)-arc transitive, and ΓB 6∼= Kv+1. Theorem 7.0.2(b) shows that

the construction of the graphs in F can be reduced to that of (G, 2)-arc transitive

near n-gonal graphs (Σ, E) of valency v ≥ 3 such that E is a G-orbit on n-cycles of

Σ, for an even integer n ≥ 4. In view of Theorem 7.5.1 above, in constructing the

graphs in F we can start, at least theoretically, from (G, 2)-arc transitive graphs

Σ. To make this approach effective, we need to know when such a graph Σ is a

near-polygonal graph (Σ, E) with E as above. The purpose of this section is to give

the following necessary and sufficient conditions for a 2-arc transitive graph to be a

near-polygonal graph.

Theorem 7.5.2 Suppose that Σ is a connected (G, 2)-arc transitive graph with

girth(Σ) ≥ 4, where G ≤ Aut(Σ). Let (σ, τ, ε) be a 2-arc of Σ and set H = Gστε.

Then the following conditions (a)-(c) are equivalent:

(a) there exist an integer n ≥ 4 and a G-orbit E on n-cycles of Σ such that (Σ, E)

is a near n-gonal graph;

(b) H fixes at least one vertex in Σ(ε) \ τ;

(c) there exists g ∈ NG(H) such that (σ, τ)g = (τ, ε).

Proof (a) ⇒ (b) Suppose that (Σ, E) is a near n-gonal graph for a G-orbit E

on n-cycles of Σ, where n ≥ 4. Let C(σ, τ, ε) = (σ, τ, ε, η, . . . , σ) be the basic cycle

containing the 2-arc (σ, τ, ε). Then we have η ∈ Σ(ε) \ τ. We claim that η is

fixed by H . Suppose otherwise, say ηg 6= η for some g ∈ H , then (C(σ, τ, ε))g =

(σ, τ, ε, ηg, . . . , σ) is a basic cycle containing (σ, τ, ε) which is different from C(σ, τ, ε).

This contradicts with the uniqueness of the basic cycle containing a given 2-arc, and

hence (b) holds.

(b) ⇒ (c) Suppose H fixes η ∈ Σ(ε) \ τ. Then we have H ≤ Gτεη. Since Σ

is (G, 2)-arc transitive, there exists g ∈ G such that (σ, τ, ε)g = (τ, ε, η) and hence

Gτεη = Hg. Therefore, Hg = H and g ∈ NG(H).

96 Almost covers

(c) ⇒ (a) Suppose that there exists g ∈ NG(H) such that (σ, τ)g = (τ, ε). Set

η := εg. Then η ∈ Σ(ε) \ τ, (σ, τ, ε)g = (τ, ε, η) and hence Gτεη = Hg = H . Set

σ0 = σ, σ1 = τ, σ2 = ε and σ3 = η, and set σ4 = σg3 . Then σ4 ∈ Σ(σ3) \ σ2 and

Gσ2σ3σ4 = (Gσ1σ2σ3)g = Hg = H . Now set σ5 = σg4 , then similarly σ5 ∈ Σ(σ4) \ σ3

and Gσ3σ4σ5 = (Gσ2σ3σ4)g = Hg = H . Continuing this process, we get inductively a

sequence σ0, σ1, σ2, σ3, σ4, σ5, . . . of vertices of Σ with the following properties:

(1) σi = σgi−1 for all i ≥ 1, and hence σi+1 ∈ Σ(σi) \ σi−1 for i ≥ 1 and σi = σgi

0

for i ≥ 0; and

(2) Gσi−1σiσi+1= H for all i ≥ 1.

Since we have finitely many vertices in Σ, this sequence will eventually contain re-

peated terms. Suppose σn is the first vertex in this sequence which coincides with

one of the preceding vertices. Without loss of generality we may suppose that σn co-

incides with σ0 for if σn = σi for some i ≥ 1 then we can begin with σi and relabel the

vertices in the sequence. Thus, we get an n-cycle J := (σ0, σ1, σ2, σ3, σ4, . . . , σn−1, σ0)

(note that n ≥ 4 as girth(Σ) ≥ 4). Let E denote the G-orbit on n-cycles of Σ con-

taining J . In the following we will prove that each 2-arc of Σ is contained in exactly

one of the “basic cycles” in E and hence (Σ, E) is indeed a near n-gonal graph.

By the (G, 2)-arc transitivity of Σ, it is clear that each 2-arc (σ′, τ ′, ε′) of Σ is

contained in at least one member Jx of E , where x ∈ G is such that (σ′, τ ′, ε′) =

(σ, τ, ε)x. So it suffices to show that if two members of E have a 2-arc in common

then they are identical; or, equivalently, if Jx and J have a 2-arc in common then

they are identical.

Suppose then that Jx and J have a 2-arc in common for some x ∈ G. Note that,

for each i ≥ 0, gi maps each vertex σj to σj+i and so 〈g〉 leaves J invariant (subscripts

modulo n here and in the rest of this proof). So, replacing Jx by Jxgifor some i if

necessary, we may suppose without loss of generality that (σ0, σ1, σ2) is a common

2-arc of Jx and J . Then (σ0, σ1, σ2) ∈ Jx implies that (σ0, σ1, σ2) = (σi−1, σi, σi+1)x

for some 1 ≤ i ≤ n. Thus, (σ0, σ1, σ2) = (σ0, σ1, σ2)gi−1x and hence gi−1x ∈ H . From

the properties (1)-(2) above, we then have σxj+i−1 = σgi−1xj = σj for each vertex σj

on J . That is, σxj = σj−i+1 for each j and hence Jx = J . Thus, we have proved

that each 2-arc of Σ is contained in exactly one member of E , and so (Σ, E) is a near

n-gonal graph. 2

Chapter 8

Flag graphs: A generalconstruction

He who learns but does not think is lost; he who thinks but does

not learn is in great danger.


In this chapter we temporarily leave the case where v = k + 1 ≥ 3. Instead we

will give a natural construction of a large class of symmetric graphs, namely the class

of G-symmetric graphs Γ such that the dual 1-design of D(B) contains no repeated

blocks. We will prove that up to isomorphism this construction produces all such

graphs, and in particular that Γ can be reconstructed from the quotient ΓB and

the action of G on B. The study in this chapter reveals a close connection between

symmetric graphs and 1-designs. In fact, the ingredients for our construction are a

G-point-transitive and G-block-transitive 1-design D, a G-orbit Θ on the flags of D

satisfying some natural conditions, and a certain self-pairedG-orbit Ψ on the ordered

pairs of distinct flags of D. Given these, the constructed graph, called a flag graph, is

defined to have vertex set Θ and arc set Ψ. The construction was initially introduced

in the course of our attempt to construct symmetric graphs with v = k + 1 ≥ 3.

However, for convenience of narration we will first present in this chapter the general

construction. The utility of this construction is not fully explored in this thesis: We

just apply it to characterizing two large classes of symmetric graphs, namely the

classes of symmetric graphs with k = 1 and v = k + 1 ≥ 3, respectively. It is hoped

that some interesting symmetric graphs can be constructed and characterized by

98 Flag graphs

using this rather general approach.

8.1 Preliminary

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition B. We

use D∗(B) := (ΓB(B), B, I∗) to denote the dual 1-design of D(B) = (B,ΓB(B), I).

So the “points” of D∗(B) are those blocks of B which are adjacent to B, and the

“blocks” of D∗(B) are the points of B. Note that the trace of the “block” α ∈ B

of D∗(B) is the subset ΓB(α) of ΓB(B), and that D∗(B) has block size r, where

r = |ΓB(α)| as in Section 3.2. By Lemma 3.2.5(b), GB induces a point-, block-

and flag-transitive group of automorphisms of D∗(B). As a vital observation, we

notice that D∗(B) can be “expanded” to the following 1-design which has point set

B and admits G as a point- and block-transitive group of automorphisms. For each

α ∈ V (Γ), set

L(α) := B(α) ∪ ΓB(α). (8.1)

We should warn that, for distinct vertices α, β of Γ, it may happen that L(α) = L(β).

(This might be true even for distinct vertices α, β in the same block of B.) Denote

by L the set of all L(α), α ∈ V (Γ), with repeated ones identified. Note that L(α) =

L(β) if and only if L(αg) = L(βg) for any g ∈ G. Therefore, (L(α))g := L(αg), for

α ∈ V (Γ) and g ∈ G, defines an action of G on L. We define

D(Γ,B) := (B,L)

to be the incidence structure where B is incident with L(α) if and only if B ∈ L(α).

Lemma 8.1.1 Suppose that Γ is a G-symmetric graph admitting a nontrivial G-

invariant partition B. Then

(a) D(Γ,B) is a 1-design with block size r + 1; and

(b) D(Γ,B) admits G as a group of automorphisms, and G is transitive on the

points and the blocks of D(Γ,B).

Proof It is clear that G is transitive on B and on L, and that G preserves the

incidence relation of D(Γ,B). So G induces a group of automorphisms of D(Γ,B),

and each B ∈ B is incident with the same number of elements of L. Clearly, D(Γ,B)

has block size r + 1. 2

Preliminary 99

We notice that in a lot of cases (see Examples 8.1.1 and 8.1.2 below) the 1-

design D∗(B) contains no repeated blocks, and the purpose of this chapter is to

give a construction of all G-symmetric graphs with this property. (Recall that two

“blocks” α, β of D∗(B) are said to be repeated if ΓB(α) = ΓB(β).) The feasibility of

such a construction lies on the observation that in this case the flags (B(α),L(α))

of D(Γ,B), for α ∈ V (Γ), are pairwise distinct, or equivalently, for each B ∈ B the

members of

L(B) := L(α) : α ∈ B

are pairwise distinct. Therefore, in this case V (Γ) can be identified with the set

Θ(Γ,B) := (B(α),L(α)) : α ∈ V (Γ)

of flags of D(Γ,B) by α 7→ (B(α),L(α)). We denote by

E(B) := ΓB(α) : α ∈ B (8.2)

the set of distinct traces of the “blocks” of D∗(B). Denote by GB,ΓB(α) and GB,L(α)

the setwise stabilizers of ΓB(α) and L(α) inGB, respectively. Note that GB preserves

L(B) and hence induces an action on L(B).


invariant partition B. Then Θ(Γ,B) is a G-orbit on the flags of D(Γ,B). The flags

(B(α),L(α)) for α ∈ V (Γ) are pairwise distinct if and only if D∗(B) contains no

repeated blocks, and in this case the following (a)-(d) hold (where B ∈ B in (c) and

(d)).

(a) The mapping ρ : α 7→ (B(α),L(α)), for α ∈ V (Γ), defines a bijection from

V (Γ) to Θ(Γ,B).

(b) The actions of G on V (Γ) and on Θ(Γ,B) are permutationally equivalent

with respect to the bijection ρ in (a).

(c) The action of GB on B is permutationally equivalent to the actions of GB

on E(B), L(B) with respect to the bijections defined by α 7→ ΓB(α), α 7→ L(α), for

α ∈ B, respectively. Hence we have GB,ΓB(α) = GB,L(α) = Gα.

(d) GB,L(α) is transitive on ΓB(α), for α ∈ B.

100 Flag graphs

Proof Since G is transitive on V (Γ), it is easy to see that Θ(Γ,B) is a G-orbit

on the flags of D(Γ,B). Clearly, the flags (B(α),L(α)), (B(β),L(β)) in Θ(Γ,B)

corresponding to two distinct vertices α, β are identical if and only if B(α) = B(β)

and L(α) = L(β), that is, if and only if α, β are in the same block of B and

ΓB(α) = ΓB(β). In other words, the flags (B(α),L(α)) for α ∈ V (Γ) are pairwise

distinct if and only if D∗(B) contains no repeated blocks. In this case it is easily

checked that both (a) and (b) are true. The truth of (c) follows from a routine

argument. From Lemma 3.2.5(c) it follows that Gα is transitive on ΓB(α), that is,

GB,L(α) is transitive on ΓB(α). Thus (d) is proved. 2

Remark 8.1.1 Under the assumption that D∗(B) contains no repeated blocks,

D(Γ,B) is an extension of D∗(B) if and only if ΓB is a complete graph. (See Section

2.3 for the definition of an extension of a design.) In this case D(Γ,B) is a 2-design

with G acting doubly transitively on its points.

We conclude this section by giving examples of two large classes of symmetric

graphs such that D∗(B) contains no repeated blocks.

Example 8.1.1 Suppose Γ is a G-symmetric graph such that k = 1, that is,

Γ[B,C] ∼= K2 for adjacent blocks B,C of B. Then clearly we have ΓB(α)∩ΓB(β) = ∅

for distinct vertices α, β in the same block of B. In particular, D∗(B) contains no

repeated blocks. Note that in this case the block size r of D∗(B) is equal to the

valency val(Γ) of Γ. Moreover, we have val(ΓB) = vr.

We remind the reader that symmetric graphs satisfying k = 1 have appeared in

Sections 4.1, 4.2 and Theorem 5.1.3(a)(b).

Example 8.1.2 Suppose Γ is a G-symmetric graph such that k < v and GB is

doubly transitive on B. Then D∗(B) must contain no repeated blocks. In fact,

suppose otherwise, then since GB is doubly transitive on the blocks of D∗(B) we

would have ΓB(α) = ΓB(β) for all α, β ∈ B. This implies k = v and thus contradicts

our assumption.

Flag Graphs 101

8.2 Flag graphs

For simplicity we assume without mentioning explicitly that the 1-designs used for

our constructions in this chapter have no repeated blocks. Let D = (V,B) be a 1-

design. As usual we may identify each block L ∈ B with the subset of V consisting of

the points incident with L. Let Θ be a set of flags of D, and Ψ a subset of the set Θ(2)

of ordered pairs of distinct flags in Θ. If Ψ is self-paired, that is, ((σ, L), (τ,N)) ∈ Ψ

implies ((τ,N), (σ, L)) ∈ Ψ, then we define the flag graph of D with respect to (Θ,Ψ),

denoted by Γ(D,Θ,Ψ), to be the graph with vertex set Θ in which two “vertices”

(σ, L), (τ,N) ∈ Θ are adjacent if and only if ((σ, L), (τ,N)) ∈ Ψ. The self-parity of

Ψ guarantees that this graph is well-defined. For a given point σ of D, we denote by

Θ(σ) the set of flags in Θ with point entry σ. Let G be a group of automorphisms

of D. If Θ is a G-orbit on the flags of D, then Θ(σ) is a Gσ-orbit on the flags of D

with point entry σ. In this case, Γ(D,Θ,Ψ) is G-vertex-transitive and its vertex set

Θ admits

B(Θ) := Θ(σ) : σ ∈ V (8.3)

as a natural G-invariant partition. If furthermore Ψ is a G-orbit on Θ(2) (under the

induced action), then Γ(D,Θ,Ψ) is G-symmetric. For a flag (σ, L) of D, we use Gσ,L

to denote the subgroup of G fixing (σ, L), that is, the subgroup of G fixing σ and

L setwise. For the purpose of this chapter, the G-orbit Θ will be required to satisfy

some additional properties.

Definition 8.2.1 Let D be a G-point-transitive and G-block-transitive 1-design

with block size at least 2. Let σ be a point of D. A G-orbit Θ on the flags of D is

said to be feasible if

(a) |Θ(σ)| ≥ 2; and

(b) Gσ,L is transitive on L \ σ, for some (and hence all) (σ, L) ∈ Θ.

102 Flag graphs

N

L

Figure 5 A compatible ordered pair of flags

Since G is transitive on the points of D, the validity of (a), (b) above does not

depend on the choice of the point σ. Let Θ be a feasible G-orbit on the flags of D.

We say that ((σ, L), (τ,N)) ∈ Θ(2) is compatible with Θ if σ 6= τ and σ, τ ∈ L ∩ N

(see Figure 5). In the following we use C(D,Θ) to denote the set of those members of

Θ(2) which are compatible with Θ. One can easily see that C(D,Θ) is a G-invariant

subset of Θ(2). In this chapter we will consider only those flag graphs Γ(D,Θ,Ψ)

such that D and G are as in Definition 8.2.1, Θ is a feasible G-orbit on the flags of

D, and Ψ is a self-paired G-orbit on C(D,Θ) (that is, Ψ is a self-paired G-orbit on

Θ(2) whose members are all compatible with Θ). For such a Ψ, either L = N for all

((σ, L), (τ,N)) ∈ Ψ, or L 6= N for all ((σ, L), (τ,N)) ∈ Ψ. For convenience we will

call such a graph the G-flag graph of D with respect to (Θ,Ψ). In the following we

show that these graphs can represent all G-symmetric graphs admitting a nontrivial

G-invariant partition B such that D∗(B) contains no repeated blocks.


invariant partition B such that D∗(B) contains no repeated blocks. Let r be the block

size of D∗(B), that is, r = |ΓB(α)|. Then Γ ∼= Γ(D,Θ,Ψ) for a certain G-point-

transitive and G-block-transitive 1-design D with block size r + 1, a certain feasible

G-orbit Θ on the flags of D, and a certain self-paired G-orbit Ψ on C(D,Θ).

Conversely, for any G-point-transitive and G-block-transitive 1-design D with no

repeated blocks and with block size r + 1, any feasible G-orbit Θ on the flags of D,

Flag Graphs 103

and any self-paired G-orbit Ψ on C(D,Θ), the graph Γ := Γ(D,Θ,Ψ), group G,

partition B := B(Θ) and integer r satisfy all the conditions above.

Proof Suppose that Γ, G, B and r are as in the first part of the theorem. Then by

Lemma 8.1.1, D := D(Γ,B) is a G-point-transitive and G-block-transitive 1-design

with block size r + 1. From Lemma 8.1.2, Θ := Θ(Γ,B) is a G-orbit on the flags of

D, and the mapping ρ : γ 7→ (B(γ),L(γ)), for γ ∈ V (Γ), is a bijection from V (Γ)

to Θ. In particular, we have |Θ(B)| = |B| ≥ 2. For (B,L) ∈ Θ(B), say L = L(α)

for some α ∈ B, we have L \ B = ΓB(α). So it follows from Lemma 8.1.2(d) that

GB,L is transitive on L \ B. Therefore, Θ is a feasible G-orbit on the flags of D.

Clearly, for each arc (α, β) of Γ, we have B(α) 6= B(β) and B(α), B(β) ∈ L(α)∩

L(β). Therefore, setting

Ψ := ((B(α),L(α)), (B(β),L(β))) : (α, β) ∈ Arc(Γ),

then Ψ ⊆ C(D,Θ) and Ψ is self-paired. By Lemma 8.1.2(b), the actions of G on

V (Γ) and Θ are permutationally equivalent with respect to the bijection ρ defined

above. Since Γ is G-symmetric, this implies that Ψ = ((B(α),L(α)), (B(β),L(β)))G,

for a fixed arc (α, β) of Γ. Hence Ψ is a self-paired G-orbit on C(D,Θ). One can

easily check that the bijection ρ defines an isomorphism from Γ to the G-flag graph

Γ(D,Θ,Ψ), and hence the first part of Theorem 8.2.1 is proved.

Suppose conversely that D, G,Θ,Ψ and r are as in the second part of Theorem

8.2.1. Let Γ := Γ(D,Θ,Ψ), and let B := B(Θ) be as defined in (8.3). Then it follows

from the definition that Γ is a G-symmetric graph with vertex set Θ, and that B

is a nontrivial G-invariant partition of Θ with block size |Θ(σ)| ≥ 2, where σ is a

point of D. To complete the proof, we need to show that the block size of D∗(Θ(σ))

is equal to r and that D∗(Θ(σ)) contains no repeated blocks.

Let Θ(σ),Θ(τ) be adjacent blocks of B. Then there exist (σ, L) ∈ Θ(σ) and

(τ,N) ∈ Θ(τ) such that (σ, L), (τ,N) are adjacent in Γ, that is, ((σ, L), (τ,N)) ∈ Ψ.

So we have σ 6= τ and σ, τ ∈ L∩N by the compatibility of the members of Ψ. Since Θ

is feasible, it follows from (b) in Definition 8.2.1 that, for any τ1 ∈ L\σ, there exists

g ∈ Gσ,L such that τ g = τ1. Setting N1 := Ng, then we have (τ1, N1) = (τ,N)g ∈ Θ.

Since g fixes σ, it fixes Θ(σ) setwise, and moreover σ ∈ N implies σ ∈ N1. Also, σ 6=

τ implies that σ = σg 6= τ g = τ1. Thus we have ((σ, L), (τ1, N1)) = ((σ, L), (τ,N))g ∈

104 Flag graphs

Ψ, that is, (σ, L) and (τ1, N1) are adjacent in Γ. Hence Θ(τ1) ∈ ΓB((σ, L)) provided

that τ1 ∈ L \ σ. We now prove that the converse of this is true as well. In fact,

suppose that Θ(δ) ∈ ΓB((σ, L)). Then there exists (δ,M) ∈ Θ(δ) such that (σ, L)

and (δ,M) are adjacent in Γ. So ((σ, L), (δ,M)) ∈ Ψ and hence there exists h ∈ G

such that ((σ, L), (τ,N))h = ((σ, L), (δ,M)). Thus we have h ∈ Gσ,L, τh = δ and

Nh = M . Since h fixes σ and fixes L setwise, and since τ ∈ L \ σ, we have

δ = τh ∈ L \ σ. So we have proved that ΓB((σ, L)) = Θ(δ) : δ ∈ L \ σ, and

thus D∗(Θ(σ)) has block size |L \ σ| = r. Moreover, since D contains no repeated

blocks, we have L 6= L1 for distinct (σ, L), (σ, L1) ∈ Θ(σ). This together with the

argument above implies that ΓB((σ, L)) 6= ΓB((σ, L1)), and hence D∗(Θ(σ)) contains

no repeated blocks. 2

The special case where in addition ΓB is a complete graph (that is, ΓB∼= Kb+1)

is particularly interesting. Since ΓB is G-symmetric, this case occurs if and only if G

is doubly transitive on B. So in this case D(Γ,B) = (B,L) is a G-doubly transitive

and G-block-transitive 2-(b+ 1, r+ 1, λ) design, for some integer λ ≥ 1. Conversely,

if D is a G-doubly transitive and G-block-transitive 2-(b + 1, r + 1, λ) design, then

for any G-flag graph Γ := Γ(D,Θ,Ψ) of D we have ΓB(Θ)∼= Kb+1. So Theorem 8.2.1

implies the following corollary.

Corollary 8.2.1 Let b ≥ 2 and r ≥ 1 be integers, and let G be a group. Then the

following (a), (b) are equivalent.

(a) Γ is a G-symmetric graph admitting a nontrivial G-invariant partition B

such that D∗(B) contains no repeated blocks and has block size r, and such that

ΓB∼= Kb+1.

(b) Γ ∼= Γ(D,Θ,Ψ), for a G-doubly transitive and G-block-transitive 2-(b+1, r+

1, λ) design D, a feasible G-orbit Θ on the flags of D, and a self-paired G-orbit Ψ

on C(D,Θ).

Let us consider the graphs Γ in Example 8.1.2 with the additional property that

ΓB∼= Kb+1. From the corollary above, they are all G-flag graphs of some G-doubly

transitive 2-designs. So all the graphs appearing in [45, Theorems 1.1 and 1.2] are in

fact G-flag graphs. Thus, from the general theory above, the close connection of such

graphs with certain doubly transitive 2-designs shown in [45] is not a coincidence.

The Case k = 1 105

8.3 Symmetric graphs with k = 1

In this section we will study G-symmetric graphs Γ admitting a nontrivial G-

invariant partition B such that k = 1, that is, Γ[B,C] ∼= K2 for adjacent blocks

B,C of B. This seemingly trivial case is notoriously difficult to manage, even in the

case where in addition ΓB is a complete graph (see [43, Section 4]). The behaviour

of such graphs seems to be quite wild, and to the best knowledge of the author there

is no useful description of them up to now. In Example 8.1.1 we have shown that in

this case D∗(B) has block size val(Γ) and contains no repeated blocks. Hence, from

Theorem 8.2.1, Γ is isomorphic to a G-flag graph of D := D(Γ,B). In the following

we will further characterize Γ as a G-flag graph Γ(D,Θ,Ψ) with Θ satisfying some

additional condition. Using the notation in Section 8.1, we see that in this case

D∗(B) has “blocks” ΓB(α) = C ∈ B : Γ(C)∩B(α) = α, for α ∈ B. Recall that

D∗(B) has “block” set E(B) = ΓB(α) : α ∈ B.


invariant partition B such that Γ[B,C] ∼= K2 for adjacent blocks B,C of B, and let

r be the valency of Γ. Then the following (a)-(c) hold.

(a) E(B) is a GB-invariant partition of ΓB(B).

(b) If G is faithful on V (Γ), then the induced action of G on B is faithful.

(c) If L(α) = L(β) holds for some pair of distinct vertices α, β of Γ, then Γ ∼=

n ·Kr+1 for some integer n; and in this case L(γ) = L(δ) holds for any vertices γ, δ

in the same component of Γ.

Proof (a) Since there is only one edge of Γ between two adjacent blocks of B,

it follows from the definition that E(B) is a partition of ΓB(B). Suppose that

(ΓB(α))g ∩ ΓB(β) 6= ∅ for some α, β ∈ B and g ∈ GB, say Cg = D for some

C ∈ ΓB(α) and D ∈ ΓB(β). Since α is the unique vertex in B adjacent to a vertex in

C and since β is the unique vertex in B adjacent to a vertex in D, Cg = D implies

αg = β and hence (ΓB(α))g = ΓB(β). Therefore, E(B) is a GB-invariant partition

of ΓB(B).

(b) Suppose that g ∈ G fixes setwise each block of B. Then, for each B ∈ B and

α ∈ B, g fixes in particular each of the blocks in ΓB(α). So it follows from (a) that

g fixes each vertex in B. Since this holds for each B ∈ B, g fixes each vertex of Γ.

So, if G is faithful on V (Γ), then g = 1 and hence G is faithful on B.

106 Flag graphs

(c) Suppose that L(α) = L(β) for two distinct vertices α ∈ B and β ∈ C. Then

B 6= C, C ∈ ΓB(α) and B ∈ ΓB(β), and in particular B,C are adjacent. Moreover,

since there is only one edge between B and C, α, β must be adjacent in Γ. So the

transitivity of Gα on Γ(α) implies that, for each γ ∈ Γ(α), there exists g ∈ Gα such

that βg = γ. Since L(α) = L(β) we then have L(α) = (L(α))g = (L(β))g = L(γ). In

particular this implies that each block in L(α) other than B(γ) contains a (unique)

neighbour of γ, and so any two blocks in L(α) are adjacent. For distinct vertices

γ, δ ∈ Γ(α), say δ ∈ D, let δ′ be the neighbour of γ in the block D. Then by

the G-symmetry of Γ there exists h ∈ G such that (α, δ)h = (γ, δ′). This implies

(L(α))h = L(γ) and (L(δ))h = L(δ′). Since L(α) = L(δ) as shown above, we have

L(δ′) = L(γ) = L(α). Thus δ′ is adjacent to a vertex in B. However, our assumption

on Γ[B,D] implies that δ is the unique vertex in D adjacent to a vertex in B. So we

must have δ′ = δ. Thus we have shown that any two vertices in Γ(α) are adjacent.

Hence α ∪ Γ(α) induces the complete graph Kr+1, which must be a connected

component of Γ since Γ has valency r. Therefore, Γ is a union of disjoint copies of

Kr+1. Obviously in this case L(γ) = L(δ) holds for any vertices γ, δ in the same

component of Γ. 2

Part (c) of Lemma 8.3.1 implies that, if k = 1 and Γ is not a union of complete

graphs, then the sets L(α) (for α ∈ V (Γ)) of blocks of B are pairwise distinct and

thus D(Γ,B) = (B, L(α) : α ∈ V (Γ)). On the other hand, we will see in Example

8.3.1 that the opposite case can occur, that is, it may happen that L(α) = L(β)

for some pair of distinct vertices α, β of Γ. Inspired by part (a) of Lemma 8.3.1, we

give the following definition.

Definition 8.3.1 Let D and G be as in Definition 8.2.1. A G-orbit Θ on the flags

of D is said to be a 1-feasible G-orbit if it is feasible and L ∩ N = σ holds for

distinct (σ, L), (σ,N) ∈ Θ(σ), where σ is a point of D.

Now we prove that, up to isomorphism, the class of G-symmetric graphs with

k = 1 is precisely the class of G-flag graphs Γ(D,Θ,Ψ) such that Θ is 1-feasible.


invariant partition B such that Γ[B,C] ∼= K2 for adjacent blocks B,C of B, and let

r be the valency of Γ. Then Γ ∼= Γ(D,Θ,Ψ) holds for a certain G-point-transitive

The Case k = 1 107

and G-block-transitive 1-design D with block size r + 1, a certain 1-feasible G-orbit

Θ on the flags of D, and a certain self-paired G-orbit Ψ on C(D,Θ).

Conversely, for any G-point-transitive and G-block-transitive 1-design D with

block size r + 1, any 1-feasible G-orbit Θ on the flags of D, and any self-paired G-

orbit Ψ on C(D,Θ), the graph Γ := Γ(D,Θ,Ψ), group G, partition B := B(Θ) and

integer r satisfy all the conditions above.

We will show further that, in both parts of this theorem, G is faithful on the

vertices of Γ if and only if it is faithful on the points of D.

Proof For the first part, we have seen in Example 8.1.1 that D∗(B) contains no

repeated blocks and that the block size of D∗(B) is equal to r, the valency of Γ. By

Lemma 8.1.1, D := D(Γ,B) is a G-point-transitive and G-block-transitive 1-design

with block size r+1. We have shown in the proof of Theorem 8.2.1 that Θ := Θ(Γ,B)

is a feasible G-orbit on the flags of D, that Ψ := ((B(α),L(α)), (B(β),L(β))) :

(α, β) ∈ Arc(Γ) is a self-paired G-orbit on C(D,Θ), and that Γ ∼= Γ(D,Θ,Ψ).

From Lemma 8.3.1(a), we have L ∩ N = B for distinct (B,L), (B,N ) ∈ Θ(B).

Hence Θ is 1-feasible, and the first part of the theorem is proved. Moreover, by

Lemma 8.3.1(b), if G is faithful on V (Γ), then it is also faithful on the point set B

of D.

Suppose conversely that D, G, Θ, Ψ and r are as in the second part of the

theorem. We have proved in Theorem 8.2.1 that Γ := Γ(D,Θ,Ψ) is a G-symmetric

graph, that B := B(Θ) is a nontrivial G-invariant partition of the vertex set Θ of Γ,

and that D∗(Θ(σ)) has block size r and contains no repeated blocks, where σ is a

point of D. Let Θ(σ), Θ(τ) be adjacent blocks of B. Then there exist (σ, L) ∈ Θ(σ)

and (τ,N) ∈ Θ(τ) such that ((σ, L), (τ,N)) ∈ Ψ. So we have σ 6= τ and σ, τ ∈

L ∩N . Since Θ is 1-feasible this implies that, for any (σ, L1) ∈ Θ(σ) \ (σ, L) and

(τ,N1) ∈ Θ(τ)\(τ,N), we have σ 6∈ N1 and τ 6∈ L1. Thus none of ((σ, L), (τ,N1)),

((σ, L1), (τ,N)) and ((σ, L1), (τ,N1)) belongs to Ψ. In other words, the edge of Γ

joining (σ, L) and (τ,N) is the only edge between Θ(σ) and Θ(τ). Hence we have

Γ[Θ(σ),Θ(τ)] ∼= K2, and consequently the valency of Γ is equal to the block size r

of D∗(Θ(σ)). If an element of G fixes each flag in Θ, then it must fix each point of

D. So if G is faithful on the points of D, then it must be faithful on Θ, the vertex

set of Γ. This completes the proof of Theorem 8.3.1, and that of the statement

108 Flag graphs

immediately following it. 2

Analogous to Corollary 8.2.1, we have the following consequence of Theorem

8.3.1.

Corollary 8.3.1 Let v ≥ 2 and r ≥ 1 be integers, and let G be a group. Then the

following (a), (b) are equivalent.

(a) Γ is a G-symmetric graph of valency r which admits a nontrivial G-invariant

partition B of block size v such that Γ[B,C] ∼= K2 for any two blocks B,C of B (so

ΓB∼= Kvr+1).

(b) Γ ∼= Γ(D,Θ,Ψ), for a G-doubly transitive and G-block-transitive 2-(vr+1, r+

1, λ) design D, a 1-feasible G-orbit Θ on the flags of D, and a self-paired G-orbit Ψ

on C(D,Θ).

We conclude this chapter by giving the following illustrative examples.

Example 8.3.1 (a) If D is a G-flag-transitive linear space, then of course the flag

set Θ of D is the only G-orbit on the flags of D. Clearly, Θ satisfies (a) in Definition

8.2.1 and the condition in Definition 8.3.1. So Θ is feasible if and only if it satisfies

(b) in Definition 8.2.1, and in this case Θ is 1-feasible. For such a Θ, one can see

that any self-paired G-orbit on C(D,Θ) has the form Ψ = ((σ, Lστ ), (τ, Lστ )) :

(σ, τ) ∈ ∆, for some self-paired G-orbit ∆ on ordered pairs of distinct points of

D, where Lστ denotes the unique line of D through σ and τ . For such a Ψ, we set

Γ := Γ(D,Θ,Ψ) and L := Lστ for a fixed (σ, τ) ∈ ∆. From (b) in Definition 8.2.1,

for any δ ∈ L \ σ there exists g ∈ Gσ,L such that τ g = δ. So (σ, δ) = (σ, τ)g ∈ ∆

and ((σ, L), (δ, L)) = ((σ, L), (τ, L))g ∈ Ψ. It follows that (σ, L) is adjacent in Γ to

any (δ, L) ∈ Θ with δ ∈ L \ σ. Therefore, each connected component of Γ is a

complete graph induced by a line Lστ , for (σ, τ) ∈ ∆. Such a graph Γ satisfies the

condition in Lemma 8.3.1(c).

(b) In particular, if D is a G-doubly transitive linear space with vr + 1 points

and block size r + 1, then D is G-flag-transitive and its flag set Θ is 1-feasible. In

this case the only self-paired G-orbit on C(D,Θ) is

Ψ := ((σ, L), (τ, L)) : L is a line of D, σ, τ are distinct points on L.

Hence D has a unique G-flag graph Γ(D,Θ,Ψ) of which each connected component

is a complete graph induced by a line of D.

The Case k = 1 109

A 1-design D with block size 2 can be viewed as a regular graph Σ, and vice

versa, if we identify the blocks of D with the edges of Σ. The automorphism groups

of the design D and the graph Σ are the same. Moreover, under this identification

each flag (σ, L) of D, say L = σ, τ, can be identified with the arc (σ, τ) of Σ.

Hence D is G-flag-transitive if and only if Σ is G-symmetric.

Example 8.3.2 A G-flag-transitive 1-design D with block size r+1 := 2 such that

each point is incident with c ≥ 2 blocks can be identified with a G-symmetric graph

Σ of valency c. Since D is G-flag-transitive, the only G-orbit on the flags of D is the

set Θ of all flags of D, that is, the arc set Arc(Σ) of Σ. It is clear that Θ is 1-feasible,

and that the only self-paired G-orbit on C(D,Θ) is Ψ := ((σ, τ), (τ, σ)) : (σ, τ) ∈

Arc(Σ). So we get a unique G-flag graph Π := Γ(D,Θ,Ψ), which has vertex set

Arc(Σ) and edges joining (σ, τ) and (τ, σ), for all pairs σ, τ of adjacent vertices of

Σ. Clearly, we have Π ∼= n · K2 and ΠB(Θ)∼= Σ, where n is the number of edges

of Σ. From Theorem 8.3.1, these graphs Π can represent all G-symmetric graphs Γ

of valency r = 1 such that V (Γ) admits a nontrivial G-invariant partition B with

Γ[B,C] ∼= K2 for adjacent blocks B,C of B. Moreover, any G-symmetric graph Σ

with valency at least 2 can appear as the quotient ΓB of such a graph Γ.

The graph Γ in Corollary 8.3.1(a) with the additional property r = 1 is precisely

the unique G-flag graph (given in Example 8.3.1(b)) of a trivial G-doubly transitive

linear space D with v + 1 points. Corollary 8.3.1 and Examples 8.3.1(b), 8.3.2

together imply the characterization of such graphs given in [43, Theorem 4.2].

110 Flag graphs

Chapter 9

The case k = v − 1 ≥ 2:Construction

If I have presented one corner of the square and they cannot come

back to me with the other three, I should not go over the points again.


In this chapter, we continue our study of G-symmetric graphs Γ with v = k+1 ≥

3, without necessarily assuming the non-repetition of blocks of D(B). We will

first give a construction of such graphs and then prove that, up to isomorphism,

it produces all of them. In particular, if D(B) contains no repeated blocks, then

the construction gives rise to 3-arc graphs introduced in Section 5.2. Note that

v = k + 1 ≥ 3 does not guarantee the non-repetition of the blocks of the dual 1-

design of D(B). Thus in this case we cannot apply the G-flag graph construction to

Γ directly. The approach we will use is to consider the graph Γ′ defined in Definition

4.1.1, which is G-symmetric and admits the same G-invariant partition B such that

Γ′[B,C] ∼= K2 for blocks B,C of B adjacent in Γ′B (= ΓB). As in the previous

chapter, the construction here requires a G-point-transitive and G-block-transitive

1-design D with no repeated blocks, and the resultant graph is a certain flag graph of

D. The case where in addition ΓB is a complete graph occurs if and only if the design

D involved is a G-doubly transitive 2-design. Since, as a result of the classification

of finite simple groups, all the finite doubly transitive groups are known (see Section

2.1), our construction makes it possible to classify all such graphs Γ. As a moderate

goal, we will in the last two sections of this chapter classify all such graphs Γ in the

112 The case k = v − 1 ≥ 2: construction

case where D is a classic projective or affine geometry.

9.1 Preliminary discussion

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition B such

that v = k + 1 ≥ 3. As before, for each α ∈ V (Γ), we use B(α) to denote the set

of blocks of B which are adjacent to B(α) in ΓB but contain no vertex adjacent to

α in Γ, that is, B(α) = ΓB(B(α)) \ ΓB(α). In Section 4.3 we have seen that, for

B ∈ B, B(B) = B(α) : α ∈ B is a GB-invariant partition of ΓB(B). In Definition

4.1.1 we defined Γ′ to be the graph with the same vertices as Γ in which two vertices

α, β are adjacent if and only if they are mates, that is, if and only if B(α) ∈ B(β)

and B(β) ∈ B(α) hold simultaneously. Recall that Γ′ is G-symmetric (Theorem

4.1.1) and admits the same G-invariant partition B such that Γ′[B,C] ∼= K2 for

blocks B,C of B adjacent in Γ′B (= ΓB). In other words, Γ′ satisfies the condition

of Example 8.1.1, and hence the discussion in Section 8.3 applies to Γ′. Using the

notation in Section 8.1, we know that B(α) = Γ′B(α) and thus B(B) is equal to E(B)

defined in (8.2) for Γ′. Set

L′(α) := B(α) ∪ B(α).

Then L′(α) is equal to L(α) defined in (8.1) for Γ′. From Lemma 8.3.1(c), if there

exist two distinct vertices α, β such that L′(α) = L′(β), then Γ′ is a union of disjoint

copies of Km+1, where m is the multiplicity of D(B). In the following we denote by

L′ the set of all the distinct L′(α), for α ∈ V (Γ). Then, as shown in Section 8.1,

G induces a natural action on L′ defined by (L′(α))g := L′(αg) for α ∈ V (Γ) and

g ∈ G. Clearly, we have

D(Γ′,B) = (B,L′)

and

Θ(Γ′,B) = (B(α),L′(α)) : α ∈ V (Γ).

Set

L′(B) := L′(α) : α ∈ B.

From Lemma 8.1.2(c) the action of GB on B is permutationally equivalent to the

actions of GB on B(B), L′(B) with respect to the bijections defined by α 7→ B(α),

Construction 113

α 7→ L′(α), for α ∈ B, respectively. By Theorem 4.3.2(b), these actions are doubly

transitive. In the following we collect some simple results that we will use in the

next section.


invariant partition B with block size v = k + 1 ≥ 3. Let m be the multiplicity of

D(B), and let B ∈ B, α ∈ B and C ∈ B(α). Then the following (a)-(d) hold.

(a) D(Γ′,B) is a 1-design with block size m+ 1 which admits G as a point- and

block-transitive group of automorphisms.

(b) Θ(Γ′,B) is a G-orbit on the set of flags of D(Γ′,B), and the actions of G on

V (Γ) and Θ(Γ′,B) are permutationally equivalent with respect to the bijection defined

by ρ : α 7→ (B(α),L′(α)), for α ∈ V (Γ). Hence we have GB,B(α) = GB,L′(α) = Gα.

(c) GB,L′(α) is transitive on B(α), for α ∈ B.

(d) GB,C is transitive on L′(B) \ L′(α).

Proof Parts (a)-(c) follow directly from Lemmas 8.1.1, 8.1.2 and Example 8.1.1.

Since the actions of GB on B(B) and L′(B) are permutationally equivalent with

respect to the bijection defined by B(γ) 7→ L′(γ), for γ ∈ B, part (d) is a restatement

of Theorem 4.3.2(b). 2

9.2 Construction

We will use the notation and terminology of the previous chapter. Let D be a G-

point-transitive and G-block-transitive 1-design with no repeated blocks and with

block size at least 2. Let Θ be a 1-feasible G-orbit on the flags of D. We use

F(D,Θ) to denote the set of ordered pairs ((σ, L), (τ,N)) ∈ Θ(2) which are not

in C(D,Θ) but are such that there exist (σ, L′) ∈ Θ(σ) and (τ,N ′) ∈ Θ(τ) with

((σ, L′), (τ,N ′)) ∈ C(D,Θ). In other words, ((σ, L), (τ,N)) ∈ F(D,Θ) if and only if

σ 6∈ N , τ 6∈ L but σ ∈ N ′, τ ∈ L′ for some (σ, L′), (τ,N ′) ∈ Θ. In this case we have

L 6= L′, N 6= N ′, and the 1-feasibility of Θ implies that both (σ, L′) and (τ,N ′) are

unique. Moreover, for any (σ, L1), (τ,N1) ∈ Θ with L1 6= L′ and N1 6= N ′, we have

((σ, L1), (τ,N1)) ∈ F(D,Θ). It is easy to see that F(D,Θ) is a G-invariant subset of

Θ(2). For a subset Ψ of F(D,Θ), we set

Ψ′ := ((σ, L′), (τ,N ′)) : ((σ, L), (τ,N)) ∈ Ψ.


Then Ψ′ is a subset of C(D,Θ). Clearly, if Ψ is a G-orbit on F(D,Θ), then Ψ′ is

a G-orbit on C(D,Θ); and if Ψ is self-paired, then Ψ′ is self-paired as well. (The

converses of these assertions are not necessarily true.) To construct G-symmetric

graphs with v = k + 1 ≥ 3, we should impose more conditions on Θ.

Definition 9.2.1 Let D and G be as above. A G-orbit Θ on the flags of D is said

to be a strict 1-feasible G-orbit if it is 1-feasible and is such that |Θ(σ)| ≥ 3 and

that, for (σ, L) ∈ Θ and τ ∈ L \ σ, Gστ is transitive on Θ(σ) \ (σ, L).

The first major result in this chapter is the following theorem.


invariant partition B with block size v = k + 1 ≥ 3, and let m be the multiplicity

of D(B). Then Γ ∼= Γ(D,Θ,Ψ) holds for a certain G-point-transitive and G-block-

transitive 1-design with block size m+ 1, a certain strict 1-feasible G-orbit Θ on the

flags of D, and a certain self-paired G-orbit Ψ on F(D,Θ).

Conversely, for any G-point-transitive and G-block-transitive 1-design D with no

repeated blocks and with block size m+1, any strict 1-feasible G-orbit Θ on the flags

of D, and any self-paired G-orbit Ψ on F(D,Θ), the graph Γ := Γ(D,Θ,Ψ), group

G, partition B := B(Θ) and integer m satisfy all the conditions above.

Remark 9.2.1 In both parts of Theorem 9.2.1, G is faithful on the vertices of Γ

if and only if it is faithful on the points of D. Moreover, the graph Γ′ defined in

Definition 4.1.1 for Γ is isomorphic to the G-flag graph Γ(D,Θ,Ψ′).

Proof of Theorem 9.2.1 Suppose that Γ, G and B are as in the first part of the

theorem, and let Γ′ be the graph defined in Definition 4.1.1. By Lemma 9.1.1(a)(b),

D := D(Γ′,B) is a G-point-transitive and G-block-transitive 1-design with block

size m + 1, and Θ := Θ(Γ′,B) is a G-orbit on the flags of D. It follows from the

definition that Θ(B) = (B,L) : L ∈ L′(B) for B ∈ B. So |Θ(B)| = v ≥ 3 and

L ∩ N = B holds for distinct flags (B,L), (B,N ) in Θ(B). For (B,L) ∈ Θ(B),

say L = L′(α) for some α ∈ B, we have L \ B = B(α) and Θ(B) \ (B,L) =

(B,N ) : N ∈ L′(B) \ L. So Lemma 9.1.1(c)(d) and the argument above imply

that Θ is a strict 1-feasible G-orbit on the flags of D.

Construction 115

For an arc (α, β) of Γ, the blocks B := B(α) and C := B(β) are adjacent in

ΓB. So there exist α′ ∈ B and β ′ ∈ C such that α′, β ′ are mates, that is, (α′, β ′) ∈

Arc(Γ′). Thus we have B ∈ L′(β ′) and C ∈ L′(α′). It follows from the definition

that B 6∈ L′(β) and C 6∈ L′(α), and therefore we have ((B,L′(α)), (C,L′(β))) ∈

F(D,Θ). Thus, setting Ψ := ((B(α),L′(α)), (B(β),L′(β))) : (α, β) ∈ Arc(Γ),

then Ψ ⊆ F(D,Θ) and Ψ is clearly self-paired. By Lemma 9.1.1(b), the actions

of G on V (Γ) and Θ are permutationally equivalent with respect to the bijection

ρ : γ 7→ (B(γ),L′(γ)), for γ ∈ V (Γ). Since Γ is G-symmetric, this implies that

Ψ is a (self-paired) G-orbit on F(D,Θ). It is easily checked that the bijection ρ

above defines an isomorphism from Γ to Γ(D,Θ,Ψ), and hence the first part of

Theorem 9.2.1 is proved. In addition, from Theorem 4.3.1(c), if G is faithful on

the vertices of Γ, then it is also faithful on the points of D. Clearly, we have

Ψ′ = ((B(α′),L′(α′)), (B(β ′),L′(β ′))) : (α′, β ′) ∈ Arc(Γ′). From the comments

before Definition 9.2.1, Ψ′ is a self-paired G-orbit on C(D,Θ). Thus, from the proof

of Theorem 8.3.1, it follows that Γ′ ∼= Γ(D,Θ,Ψ′).

Suppose conversely that D, G,Θ,Ψ and m are as in the second part of the theo-

rem. Let Γ := Γ(D,Θ,Ψ), and let B := B(Θ) be as defined in (8.3). Then it follows

from the definition that Γ is a G-symmetric graph with vertex set Θ, and B is a

nontrivial G-invariant partition of Θ with block size v := |Θ(σ)| ≥ 3, where σ is a

point of D. To complete the proof, we need to show that the block size k of the

1-design D(Θ(σ)) induced on the block Θ(σ) of B satisfies v = k + 1, and that the

multiplicity of D(Θ(σ)) is equal to m.

Let Θ(σ),Θ(τ) be adjacent blocks of B. Then there exist (σ, L) ∈ Θ(σ) and

(τ,N) ∈ Θ(τ) such that (σ, L), (τ,N) are adjacent in Γ, that is, ((σ, L), (τ,N)) ∈

Ψ. Since Ψ is a G-orbit on F(D,Θ), we have σ 6∈ N , τ 6∈ L but there exist

(σ, L′), (τ,N ′) ∈ Θ such that σ ∈ N ′, τ ∈ L′. Clearly, for any (σ, L1) ∈ Θ(σ),

we have ((σ, L1), (τ,N′)) 6∈ F(D,Θ) and hence (τ,N ′) is not adjacent in Γ to any

vertex in Θ(σ). Similarly, (σ, L′) is not adjacent in Γ to any vertex in Θ(τ). On

the other hand, since Θ is a strict 1-feasible G-orbit and since τ ∈ L′ \ σ, it

follows from Definition 9.2.1 that Gστ is transitive on Θ(σ) \ (σ, L′). Thus, for

any (σ, L1) ∈ Θ(σ) \ (σ, L′), there exists g ∈ Gστ such that (σ, L)g = (σ, L1).

Since σ 6∈ N and g fixes σ, we have σ 6∈ Ng. But σ ∈ N ′, so we have (τ,N1) :=

(τ,N)g ∈ Θ(τ) \ (τ,N ′), and (σ, L1), (τ,N1) are adjacent in Γ. Thus each vertex


in Θ(σ) \ (σ, L′) is adjacent in Γ to at least one veretx in Θ(τ) \ (τ,N ′), that

is, Γ(Θ(τ)) ∩Θ(σ) = Θ(σ) \ (σ, L′). Hence v = k+ 1. From the comments before

Definition 9.2.1, we know that Ψ′ = ((σ, L′), (τ,N ′))G and that Ψ′ is a self-paired

G-orbit on C(D,Θ). Moreover, the argument above shows that the G-flag graph

Γ(D,Θ,Ψ′) is exactly the accompanying graph Γ′ of Γ defined in Definition 4.1.1. It

follows from Theorem 8.3.1 that Γ′ has valency m. In other words, the multiplicity

of D(Θ(σ)) is equal to m.

Finally, if an element of G fixes each flag in Θ, then it must fix each point of D.

So if G is faithful on the points of D then it must be faithful on the vertices of Γ.

This completes the proof of Theorem 9.2.1, as well as that of Remark 9.2.1. 2

From the proof above, the graph Γ = Γ(D,Θ,Ψ) in Theorem 9.2.1 coexists with

the G-flag graph Γ′ = Γ(D,Θ,Ψ′). For brevity we will call such a graph Γ(D,Θ,Ψ)

a coexisting G-flag graph. Now we illustrate our construction of such graphs by

examining an important special case. The following example shows that, in this

“simplest” case, the construction produces precisely the 3-arc graphs associated

with (G, 2)-arc transitive graphs.

Example 9.2.1 As mentioned before Example 8.3.2, a G-flag-transitive 1-design

D with block size 2 can be viewed as a G-symmetric graph Σ, and vice versa, if

we identify the blocks of D with the edges of Σ. Under this identification each flag

of D can be identified with an arc of Σ, and hence the valency v of Σ is equal to

the number of blocks of D incident with a given point. We assume v ≥ 3 in the

following. Since D is G-flag-transitive, the only G-orbit on the flags of D is the set

Θ of all flags of D, that is, the arc set Arc(Σ) of Σ. Clearly, Θ is 1-feasible and

|Θ(σ)| = v ≥ 3. The second condition in Definition 9.2.1 is equivalent to requiring

that Σ is (G, 2)-arc transitive. Therefore, D has a strict 1-feasible G-orbit on its

flags if and only if Σ is (G, 2)-arc transitive, and in this case the only such G-orbit

is Θ. The G-invariant partition B(Θ) of Θ (defined in (8.3)) can be identified with

the G-invariant partition B(Σ) := B(σ) : σ ∈ V (Σ) of Arc(Σ) defined in Section

5.2, where B(σ) is the set of arcs of Σ initiated at σ. Moreover, an ordered pair

((σ, L), (τ,N)) ∈ Θ(2), say L = σ, σ′, N = τ, τ ′, lies in F(D,Θ) if and only

if (σ′, σ, τ, τ ′) is a 3-arc of Σ. So we may identify such a pair ((σ, L), (τ,N)) with

(σ′, σ, τ, τ ′), and thus identify F(D,Θ) with Arc3(Σ). Hence a self-paired G-orbit

Doubly Transitive Designs 117

Ψ on F(D,Θ) can be identified with a self-paired G-orbit ∆ on Arc3(Σ), and vice

versa. Therefore, the flag graph Γ(D,Θ,Ψ) is isomorphic to the 3-arc graph Ξ(Σ,∆)

of Σ with respect to ∆.

Remark 9.2.2 Let Γ be a G-symmetric graph Γ admitting a nontrivial G-invariant

partition B with block size v = k+1 ≥ 3. Then D(B) contains no repeated blocks if

and only if the 1-design D := D(Γ′,B) has block size 2. In this case we may identify

D with the quotient graph ΓB by identifying each block B,C of D with the edge

of ΓB joining B and C. So Theorem 9.2.1 and the discussion in Example 9.2.1 imply

Theorem 5.2.3 as a consequence.

9.3 Coexisting G-flag graphs of doubly transitive

designs

In this section we examine the case where v = k + 1 ≥ 3 and in addition ΓB is a

complete graph, that is, ΓB∼= Kmv+1 (note that val(ΓB) = mv by Theorem 4.3.1(a)).

Similar to Corollaries 8.2.1 and 8.3.1, we have the following consequence of Theorem

9.2.1.

Corollary 9.3.1 Let v ≥ 3 and m ≥ 1 be integers. Then the following (a), (b) are

equivalent.

(a) Γ is a G-symmetric graph admitting a nontrivial G-invariant partition B of

block size v such that v = k + 1 and ΓB∼= Kmv+1.

(b) Γ ∼= Γ(D,Θ,Ψ), for a G-doubly transitive and G-block-transitive 2-(mv +

1, m + 1, λ) design D, a strict 1-feasible G-orbit Θ on the flags of D, and a self-

paired G-orbit Ψ on F(D,Θ).

Moreover, the integer m is equal to the multiplicity of the 1-design D(B), and G

is faithful on V (Γ) if and only if it is faithful on the points of D. Thus, by studying

G-doubly transitive, G-block-transitive 2-(mv + 1, m + 1, λ) designs D, it seems

feasible to classify all G-symmetric graphs Γ in Corollary 9.3.1. If and only if m = 1

such a design D is a G-doubly transitive 2-(v + 1, 2, 1) design, that is, a G-doubly

transitive trivial linear space. From Example 9.2.1, in this case the existence of

coexisting G-flag graphs of D requires that D is G-triply transitive, and such graphs


are precisely the 3-arc graphs of the (G, 2)-arc transitive graph Kv+1 and thus are

those graphs classified in Theorem 6.6.1. In the following we suppose m ≥ 2 and D

admits a strict 1-feasible G-orbit Θ on its flags. Let V denote the point set of D.

The double transitivity of G on V implies that, for each pair σ, τ of distinct points

of D, there exists (σ, L) ∈ Θ such that τ ∈ L. Thus, since Θ is 1-feasible, V \ σ

admits a Gσ-invariant partition of block size m, namely P := L\σ : (σ, L) ∈ Θ.

In particular, this implies that G is not 2-primitive and hence not 3-transitive on

V . Moreover, by the strict 1-feasibility of Θ, for any P ∈ P and τ ∈ P , Gσ,P is

transitive on P and Gστ is transitive on P \ P. This latter assertion implies that

any Gστ -orbit X on V \ P intersects with the same number of points in each of

the v − 1 blocks of P \ P, and hence v − 1 must be a divisor of |X|. (If we use

v0 = mv + 1 and k0 = m + 1 to denote the number of points and the block size of

D respectively, then this is equivalent to saying that (v0 − k0)/(k0 − 1) is a divisor

of |X|.) This necessary condition can be used to exclude some 2-transitive groups

G involved, as shown in the following example.

Example 9.3.1 (a) We show that the Higman-Sims group HS cannot serve as the

group G above. Suppose otherwise, then since HS is 2- but not 3-transitive with

degree 176, we havemv = 175 = 52×7 andm ≥ 2 by the discussion above. Hence the

only possibilities for (m, v) are (5, 35), (7, 25), (25, 7), (35, 5). For distinct σ, τ ∈ V ,

we have HSσ = PSU(3, 5) : Z2 ≥ PSU(3, 5) (split extension, see [24, pp.81]), and so

HSστ = A6.Z22 ≥ (PSU(3, 5))τ (see [24, pp.34]). In the action of PSU(3, 5) with

degree 175, (PSU(3, 5))τ has orbits of lengths 1, 21, 28, 125, respectively. So the

HSστ -orbits on V \σ, τ have lengths at least 21. In view of the necessary condition

above, this happens only when (m, v) = (25, 7) or (35, 5). In these two cases there

must have an HSστ -orbit with length 21, and either there are two remaining HSστ -

orbits with lengths 28, 125 respectively, or there is only one remaining HSστ -orbit

with length 28 + 125. Note that v − 1 is either 6 or 4, but 6 6∣

∣

∣ 28, 125, 28 + 125 and

4 6∣

∣

∣ 125, 28+125. This contradicts our condition above, and hence the group HS can

be excluded.

(b) Similarly, we can show that the Conway group Co3 cannot serve as the group

G above. Suppose otherwise, then since Co3 is 2- but not 3-transitive with degree

276, we havemv = 275 = 52×11 andm ≥ 2. Hence (m, v) = (5, 55), (11, 25), (25, 11),

Projective Flag Graphs 119

or (55, 5). By [24, pp.134], (Co3)σ = McL : Z2 ≥ McL; and by [24, pp.100]

McLτ = PSU(4, 3) has orbits of lengths 1, 22, 252 in its action of degree 275. Since

Co3 is not 3-transitive on V , (Co3)στ must have two orbits on V \ σ, τ and the

lengths of them must be 22, 252, respectively. Using the necessary condition above,

we can see that all the possibilities for (m, v) cannot appear. Hence the group Co3

can be excluded as well.

As a result of the finite simple group classification, all doubly transitive linear

spaces are known [51, Theorem 1]. Because of this, it seems possible to classify the

flag graphs Γ(D,Θ,Ψ) appeared in Corollary 9.3.1 for G-doubly transitive linear

spaces D, and this will contribute to the classification of all the graphs Γ therein.

As an effort towards this project, we will classify in the next two sections such

graphs Γ(D,Θ,Ψ) for two typical G-doubly transitive linear spaces D, namely the

projective geometry PG(d − 1, q) (d ≥ 3) and affine geometry AG(d, q) (d ≥ 2),

where G is a group with PSL(d, q) ≤ G ≤ PΓL(d, q) or AGL(d, q) ≤ G ≤ AΓL(d, q),

respectively.

Remark 9.3.1 (a) A G-doubly transitive linear space D must be G-flag-transitive,

and hence the only G-orbit on the flags of D is the flag set Θ of D. In this case Θ

satisfies (b) in Definition 8.2.1 and the condition in Definition 8.3.1 automatically.

Hence Θ is strictly 1-feasible if and only if a point is incident with at least three

lines and, for two points σ, τ , Gστ is transitive on the lines incident with σ but not τ .

Note that in this case we have F(D,Θ) = ((σ, L), (τ,N)) : (σ, L), (τ,N) ∈ Θ, σ 6∈

N, τ 6∈ L.

(b) Conversely, if the flag set of a G-flag-transitive 2-design D is 1-feasible, then

D is forced to be a linear space.

9.4 Projective flag graphs

Let d ≥ 3 be an integer, and let q = pe with p a prime and e ≥ 1. The projective

geometry PG(d− 1, q) is the geometry obtained by taking n-flats of AG(d, q) as its

(n− 1)-flats, for 1 ≤ n ≤ d. As usual in the literature we will use the same notation

to denote the (point, line)-incidence structure of PG(d− 1, q). Then, for any group

G with PSL(d, q) ≤ G ≤ PΓL(d, q), PG(d−1, q) is a G-doubly transitive linear space


with mv+1 := (qd−1)/(q−1) points in which each line contains m+1 := q+1 points

(see e.g. [84, Theorem 2.5(ii)]). So we have v = (qd−1 − 1)/(q − 1) and m = q. The

purpose of this section is to classify the coexisting G-flag graphs of PG(d−1, q), and

to characterize them as the only such graphs arising from any G-doubly transitive

2-design.

Recall that we use V (d, q) to denote the d-dimensional linear space of row vectors

over GF(q). Let V denote the point set of PG(d − 1, q). Then V = [x] : x ∈

V (d, q) \ 0, where [x] denotes the point of PG(d − 1, q) representing non-zero

multiples of the vector x. For 1 ≤ n ≤ d−1, n+1 points of PG(d−1, q) are said to

be independent [84, pp. 72] if they do not lie on any (n− 1)-flat of PG(d− 1, q). In

particular, three points of PG(d−1, q) are non-collinear if they are independent, and

collinear otherwise. We will exploit the following basic result in projective geometry.

(See [84, Theorem 2.10(iii)] for a proof in the special case where G = PGL(d, q).

The result in general case can be derived from [25, 1.4.24].)

Lemma 9.4.1 Suppose PSL(d, q) ≤ G ≤ PΓL(d, q), where d ≥ 3 and q is a prime

power. Then, for any integer n with 1 ≤ n ≤ d − 1, G is transitive on the set of

ordered (n + 1)-tuples of independent points of PG(d− 1, q).

Let Θ(P ; d, q) denote the set of flags (that is, (point, line)-flags) of PG(d− 1, q).

In the following lemma we will show that Θ(P ; d, q) is strictly 1-feasible. Thus,

setting F(P ; d, q) := F(PG(d−1, q),Θ(P ; d, q)), then from Remark 9.3.1(a) we have

F(P ; d, q) = ((σ, L), (τ,N)) : (σ, L), (τ,N) ∈ Θ(P ; d, q), σ 6∈ N, τ 6∈ L.

Two distinct lines L,N of PG(d − 1, q) are said to be intersecting if there exists a

unique point incident with both L and N (that is, L,N lie on the same plane of

PG(d− 1, q)), and skew otherwise. We use Ψ+(P ; d, q) (respectively, Ψ≃(P ; d, q)) to

denote the set of ordered pairs ((σ, L), (τ,N)) ∈ F(P ; d, q) such that L,N are inter-

secting (respectively, skew). Then Ψ+(P ; d, q) and Ψ≃(P ; d, q) consist of a partition

of F(P ; d, q). Note that Ψ≃(P ; d, q) 6= ∅ if and only if d ≥ 4 (see e.g. [84, pp.71]).

So we have F(P ; 3, q) = Ψ+(P ; 3, q).


power. Then the following (a), (b) hold.


(a) There exists a unique strict 1-feasible G-orbit on the flags of PG(d − 1, q),

namely Θ(P ; d, q).

(b) If d = 3, then G is transitive on F(P ; 3, q); if d ≥ 4, then G has two orbits

on F(P ; d, q), namely Ψ+(P ; d, q) and Ψ≃(P ; d, q).

Proof (a) Since PG(d − 1, q) is a G-doubly transitive linear space, it is G-flag-

transitive, and hence Θ(P ; d, q) is the only candidate for a strict 1-feasible G-orbit on

the flags of PG(d−1, q). In PG(d−1, q) each point is incident with (qd−1−1)/(q−1) ≥

3 lines ([84, Theorem 2.5(iii)]). For distinct points σ, τ , let L be the unique line

incident with both σ and τ . Let N1, N2 be two lines incident with σ but not τ , and

let δi ∈ Ni \σ, i = 1, 2. Then (σ, τ, δ1), (σ, τ, δ2) are triples of non-collinear points.

So by Lemma 9.4.1 there exists g ∈ G such that (σ, τ, δ1)g = (σ, τ, δ2), and hence

g ∈ Gστ . Since Ni is the unique line incident with σ and δi, this implies Ng1 = N2,

and hence Θ(P ; d, q) is strictly 1-feasible by Remark 9.3.1(a).

(b) Let ((σ1, L1), (τ1, N1)), ((σ2, L2), (τ2, N2)) ∈ Ψ+(P ; d, q). Let δi be the com-

mon point of Li and Ni, for i = 1, 2. Then (σ1, τ1, δ1), (σ2, τ2, δ2) are triples

of non-collinear points. By Lemma 9.4.1 we have (σ1, τ1, δ1)g = (σ2, τ2, δ2) for

some g ∈ G. This implies ((σ1, L1), (τ1, N1))g = ((σ2, L2), (τ2, N2)), and hence G

is transitive on Ψ+(P ; d, q). Similarly, for ((σ1, L1), (τ1, N1)), ((σ2, L2), (τ2, N2)) ∈

Ψ≃(P ; d, q), we can choose σ′i ∈ Li \ σi and τ ′i ∈ Ni \ τi, for i = 1, 2. So

(σ′1, σ1, τ1, τ

′1), (σ′

2, σ2, τ2, τ′2) are quadruples of independent points of PG(d − 1, q).

Again by Lemma 9.4.1 we have (σ′1, σ1, τ1, τ

′1)g = (σ′

2, σ2, τ2, τ′2) for some g ∈ G.

This implies ((σ1, L1), (τ1, N1))g = ((σ2, L2), (τ2, N2)), and hence G is transitive on

Ψ≃(P ; d, q). Since G preserves relative positions between lines and since Ψ+(P ; d, q)

and Ψ≃(P ; d, q) consist of a partition of F(P ; d, q), the assertions in (b) follow im-

mediately. 2

Clearly, both Ψ+(P ; d, q) and Ψ≃(P ; d, q) are self-paired. Hence the flag graphs

of PG(d−1, q) with respect to (Θ(P ; d, q),Ψ+(P ; d, q)), (Θ(P ; d, q),Ψ≃(P ; d, q)) are

well-defined. We denote these graphs by Γ+(P ; d, q), Γ≃(P ; d, q), respectively. (In

defining Γ≃(P ; d, q) we require that d ≥ 4.) From Lemma 9.4.2 they are the only

coexisting G-flag graphs of PG(d − 1, q). Moreover, we have the following charac-

terization of such graphs.



power. Suppose further that D is a 2-design, other than the trivial linear space,

which admits G as a faithful, doubly transitive group of automorphisms. Then any

coexisting G-flag graph of D is isomorphic to Γ+(P ; d, q) or Γ≃(P ; d, q).

Proof The group G has only two faithful permutation representations, namely the

natural actions on the points and hyperplanes of PG(d−1, q). Such representations

are interchangable by an outer automorphism of PΓL(d, q). So in the following it

suffices to consider the usual action of G on the point set V of PG(d− 1, q).

Since D is nontrivial, its block size is at least three. Suppose Θ is a strict 1-

feasible G-orbit on the flags of D, and let (σ, L) ∈ Θ. Then, as shown in Section

9.3, we have:

(1) N \ σ : (σ,N) ∈ Θ is a Gσ-invariant partition of V \ σ.

We claim further that:

(2) For any τ, δ ∈ L \ σ, the points σ, τ, δ must be collinear in PG(d− 1, q).

Suppose otherwise, and let ε be a point in a block N of D with (σ,N) ∈ Θ(σ)

and N 6= L. Then in PG(d − 1, q) either σ, τ, ε are non-collinear, or σ, δ, ε are

non-collinear, since otherwise σ, τ, δ would be collinear, which contradicts our as-

sumption. Without loss of generality we may suppose that σ, τ, ε are non-collinear in

PG(d−1, q). Then by Lemma 9.4.1 there exists g ∈ G such that (σ, τ, δ)g = (σ, τ, ε).

So we have g ∈ Gστ . Since g fixes τ , by (1) it must fix L setwise. On the other hand,

since g maps δ to ε, again from (1), g must map L to N . This is a contradiction

and hence (2) is proved. From this it follows that, for each (σ, L) ∈ Θ, the block L

of D consists of some collinear points of PG(d− 1, q). Moreover, we have:

(3) For each (σ, L) ∈ Θ, the block L of D is a line of PG(d− 1, q).

Suppose otherwise, then from (1), (2) there exists (σ,N1) ∈ Θ such that the

points of L and N1 lie on the same line, say L∗, of PG(d−1, q). Since d ≥ 3, we can

take (σ,N2) ∈ Θ such that the points in L and those in N2 do not lie on the same

line of PG(d− 1, q). Take a point τ ∈ L \ σ. Since Θ is strictly 1-feasible, by the

second condition in Definition 9.2.1, there exists g ∈ Gστ such that Ng1 = N2. Since

g fixes σ and τ , it must fix the line L∗ of PG(d − 1, q). Hence the points in N1 are


mapped by g to some points on L∗. That is, the points in N2 must lie on L∗. This

is a contradiction and hence (3) is proved.

The claims (1) and (3) together imply that Θ(σ) = Θ(P ; d, q)(σ) for each σ ∈ V .

So we have Θ = Θ(P ; d, q). In particular each line of PG(d − 1, q) is a block of D.

Thus it follows from the definition that F(D,Θ) = F(P ; d, q). From Lemma 9.4.2(b),

the result in Lemma 9.4.3 follows. 2

Applying Corollary 9.3.1 and Theorem 6.6.1, the discussion above leads to the

following classification theorem, which is the main result in this section.

Theorem 9.4.1 Suppose PSL(d, q) ≤ G ≤ PΓL(d, q), where d ≥ 2 and q = pe with

p a prime and e ≥ 1. Then, if and only if either d ≥ 3 or d = 2 and G is 3-

transitive, there exists a G-symmetric graph Γ with G faithful on V (Γ) which admits

a nontrivial G-invariant partition B such that v = k + 1 ≥ 3 and ΓB∼= Kmv+1,

where m is the multiplicity of D(B). Moreover, all the possibilities of such Γ, G and

the corresponding m, v can be classified as follows.

(a) Γ ∼= (q + 1) ·Kq, G = PGL(2, q).〈ψn〉 or M(n, q) (for suitable p, e and n),

and (m, v) = (1, q).

(b) (Γ, G) = (CR(q; x, n),PGL(2, q).〈ψt〉) and (m, v) = (1, q), where x ∈ GF(q)\

0, 1, n is a divisor of n(x), and t is a divisor of e with gcd(n(x), t) = n.

(c) (Γ, G) = (TCR(q; x, n),M(t/2, q)) and (m, v) = (1, q), where p is odd, e ≥ 2

is even, x ∈ GF(q)\0, 1 with n(x) even and x−1 a square of GF(q), n is an even

divisor of n(x), and t is a divisor of e with gcd(n(x), t) = n.

(d) Γ = Γ+(P ; d, q) or Γ≃(P ; d, q), where d ≥ 3, G is any doubly transitive

subgroup of PΓL(d, q), and (m, v) = (q, (qd−1 − 1)/(q − 1)). (The graph Γ≃(P ; d, q)

appears only when d ≥ 4.)

We conclude this section by proving the following properties of the projective

flag graphs Γ+(P ; d, q) and Γ≃(P ; d, q). As before, we denote by Lστ the unique line

of PG(d− 1, q) through two distinct points σ and τ .

Theorem 9.4.2 Let d ≥ 3 and q a prime power, and let Θ := Θ(P ; d, q). Then the

following (a)-(c) hold.


(a) Both Γ+(P ; d, q) and Γ≃(P ; d, q) are connected graphs with diameter two and

girth three, and with valencies (qd+1 − q3)/(q− 1) and (qd−1 − q2)(qd− q2)/(q− 1)2,

respectively.

(b) For distinct blocks Θ(σ),Θ(τ) of B(Θ), each vertex of Θ(σ) other than

(σ, Lστ ) is adjacent to exactly q vertices of Θ(τ) in Γ+(P ; d, q), and adjacent to

exactly (qd−1 − q2)/(q − 1) vertices of Θ(τ) in Γ≃(P ; d, q). In particular, for Γ :=

Γ+(P ; 3, q) we have Γ[Θ(σ),Θ(τ)] ∼= Kq,q.

(c) For PSL(d, q) ≤ G ≤ PΓL(d, q), any G-symmetric graph with vertex set Θ

(under the induced action) is isomorphic to either Γ+(P ; d, q), or Γ≃(P ; d, q), or

(qd− 1)/(q− 1) ·K(qd−1−1)/(q−1) with connected components the sets of flags incident

with a common point, or (qd−1 − 1)(qd − 1)/(q − 1)(q2 − 1) · Kq+1 with connected

components the sets of flags incident with a common line.

Proof Let (σ, L), (τ,N) ∈ Θ be distinct flags of PG(d− 1, q). If L 6= N then, since

each line of PG(d − 1, q) contains q + 1 ≥ 3 points, we can take δ ∈ L \ σ, τ, ε ∈

N\σ, τ and η ∈ Lδε\δ, ε. One can check that the sequence (σ, L), (η, Lδε), (τ,N)

is a path of Γ+(P ; d, q) with length two. In particular, if (σ, L), (τ,N) are adjacent

in Γ+(P ; d, q), then the sequence (σ, L), (η, Lδε), (τ,N), (σ, L) is a triangle. Simi-

larly, if σ 6= τ but L = N , then we can take δ ∈ L \ σ, τ and a point ε not

incident with L. Thus the sequence (σ, L), (ε, Lδε), (τ, L) is a path of Γ+(P ; d, q)

with length two. Hence Γ+(P ; d, q) is connected with diameter two and girth

three. The definition of Γ≃(P ; d, q) requires that d ≥ 4. So for any distinct

(σ, L), (τ,N) ∈ Θ, we can choose a line M which is skew with both L and N .

For any δ ∈ M , the sequence (σ, L), (δ,M), (τ,N) is a path of Γ≃(P ; d, q) with

length two. Moreover, if (σ, L), (τ,N) are adjacent in Γ≃(P ; d, q), then the sequence

(σ, L), (δ,M), (τ,N), (σ, L) is a triangle. Hence Γ≃(P ; d, q) is connected with diam-

eter two and girth three as well.

For any flag (σ, L) and any point τ not incident with L, there are exactly q lines

which are incident with τ and intersect with L at a point other than σ, namely those

lines joining τ and one of the points in L \ σ. Hence there are exactly v − q − 1

lines which are incident with τ and skew with L (note that Lστ is not skew with

L), where v = (qd−1 − 1)/(q − 1) as before. From these the assertions in (b) follow

immediately. Note that, for a point τ incident with L, (σ, L) is not adjacent to any

vertex of Ω(τ) in either Γ+(P ; d, q) or Γ≃(P ; d, q). Since L contains q + 1 points

Affine Flag Graphs 125

and PG(d − 1, q) has (qd − 1)/(q − 1) points in total, from (b) the assertion in (a)

concerning the valencies of Γ+(P ; d, q) and Γ≃(P ; d, q) follows.

Now let us prove (c). Suppose Γ is a graph with vertex set Θ which is G-

symmetric under the induced action of G on Θ. Let ((σ, L), (τ,N)) be an arc of

Γ. If σ = τ , then L 6= N , and two flags (σ1, L1), (τ1, N1) are adjacent in Γ if

and only if σ1 = τ1 and L1 6= N1. Since PG(d − 1, q) has (qd − 1)/(q − 1) points,

and since each point is incident with exactly (qd−1 − 1)/(q − 1) lines, in this case

we have Γ ∼= (qd − 1)/(q − 1) · K(qd−1−1)/(q−1). Similarly, if L = N , then we have

Γ ∼= (qd−1 − 1)(qd − 1)/(q − 1)(q2 − 1) · Kq+1. In the following we suppose that

σ 6= τ and L 6= N . Then the G-symmetry of Γ implies that there exists g ∈ G

which interchanges (σ, L) and (τ,N). So we have σ 6∈ N for otherwise we would

have σ ∈ L ∩ N and thus τ = σg ∈ (L ∩ N)g = L ∩ N , which implies σ =

τ and so contradicts with our assumption. Similarly, we have τ 6∈ L and hence

((σ, L), (τ,N)) ∈ F(P ; d, q). Thus, since Γ is G-symmetric, its arc set Arc(Γ) is a

self-paired G-orbit on F(P ; d, q). Therefore, from Lemma 9.4.2, Γ is isomorphic to

either Γ+(P ; d, q) or Γ≃(P ; d, q). 2

9.5 Affine flag graphs

For an integer d ≥ 2 and a prime power q, we use the same notation AG(d, q) to

denote the (point, line)-incidence structure of the affine geometry AG(d, q). Thus,

for any group G with AGL(d, q) ≤ G ≤ AΓL(d, q), AG(d, q) is a G-doubly transitive

linear space. The purpose of this section is to classify and characterize the coexisting

G-flag graphs of AG(d, q).

From Lemma 6.5.1 and Remark 9.3.1, it is easily verified that the flag set

Θ(A; d, q) of AG(d, q) is strictly 1-feasible. Thus, setting

F(A; d, q) := F(AG(d, q),Θ(A; d, q)),

we have

F(A; d, q) = ((σ, L), (τ,N)) : (σ, L), (τ,N) ∈ Θ(A; d, q), σ 6∈ N, τ 6∈ L.

We call two distinct lines of AG(d, q) intersecting if they share a unique common

point, parallel if they lie on the same plane but have no point in common, and skew


in the remaining case. We use Ψ+(A; d, q) (Ψ=(A; d, q), Ψ≃(A; d, q), respectively) to

denote the set of ordered pairs ((σ, L), (τ,N)) in F(A; d, q) such that L,N are inter-

secting (parallel, skew, respectively). Then Ψ+(A; d, q), Ψ=(A; d, q) and Ψ≃(A; d, q)

consist of a partition of F(A; d, q). (Note that Ψ≃(A; d, q) 6= ∅ if and only if d ≥ 3,

see [84, Theorem 1.15(i)].) Using Lemma 6.5.1 and by a similar argument as in the

proof of Lemma 9.4.2, one can prove the following lemma.


power. Then the following (a), (b) hold.

(a) There exists a unique strict 1-feasible G-orbit on the flags of AG(d, q), namely

Θ(A; d, q).

(b) If d = 2, then G has two orbits on F(A; d, q), namely Ψ+(A; 2, q) and

Ψ=(A; 2, q); if d ≥ 3, then G has three orbits on F(A; d, q), namely Ψ+(A; d, q),

Ψ=(A; d, q) and Ψ≃(A; d, q).

Clearly, Ψ+(A; d, q), Ψ=(A; d, q) and Ψ≃(A; d, q) are all self-paired. Thus the flag

graphs of AG(d, q) with respect to (Θ(A; d, q),Ψ), for Ψ = Ψ+(A; d, q), Ψ=(A; d, q)

and Ψ≃(A; d, q), are well-defined. We use Γ+(A;n, q), Γ=(A;n, q) and Γ≃(A;n, q)

respectively to denote these graphs. (In defining Γ≃(A; d, q) we require that d ≥ 3.)

From Lemma 9.5.1, these are the only coexisting G-flag graphs of AG(d, q), for

G as above. Moreover, the following lemma shows that they are in fact the only

coexisting G-flag graphs of any G-doubly transitive 2-design. The proof of this result

is similar to that of Lemma 9.4.3 and hence is omitted. (In the proof we make use of

the following fact: The only faithful permutation representation of G is its natural

action on V (d, q).)


power. Suppose further that D is a 2-design which admits G as a faithful, dou-

bly transitive group of automorphisms. Then any coexisting G-flag graph of D is

isomorphic to Γ+(A; d, q), Γ=(A; d, q), or Γ≃(A; d, q).

Remark 9.5.1 The affine geometry AG(d, q) has mv + 1 := qd points, and each

line of it contains m+1 := q points. So we have v = (qd−1)/(q−1) and m = q−1.

Thus, AG(d, q) is the trivial linear space if and only if q = 2, which in turn is

true if and only if AGL(d, q) is 3-transitive on V (d, q). Hence, from Example 9.2.1,

Affine Flag Graphs 127

Γ+(A; d, 2), Γ=(A; d, 2) and Γ≃(A; d, 2) are all 3-arc graphs of the complete graph Σ

with vertex set V (d, 2). The vertices of these three graphs are ordered pairs uw of

distinct vectors of V (d, 2). Since each plane of AG(d, 2) contains exactly four points

([84, Theorem 1.17]), one can see that uw,yz are adjacent in Γ+(A; d, 2) if and only

if w = z. So Γ+(A; d, 2) is isomorphic to 2d ·K2d−1 and is the 3-arc graph of Σ with

respect to the set of all 3-cycles of Σ. Similarly, uw,yz are adjacent in Γ=(A; d, 2)

if and only if u,w,y, z are distinct and u − w = y − z, and they are adjacent in

Γ≃(A; d, 2) if and only if u,w,y, z do not lie on the same plane of AG(d, 2). Thus

Γ=(A; d, 2) and Γ≃(A; d, 2) are, respectively, the 3-arc graphs Ξ1(d, 2) and Ξ2(d, 2)

defined in Example 6.5.1.

From Corollary 9.3.1 and the discussion above, we come to the following main

result of this section.

Theorem 9.5.1 Suppose AGL(d, q) ≤ G ≤ AΓL(d, q), where d ≥ 2 and q is a

prime power. Then there exists a G-symmetric graph Γ with G faithful on V (Γ)

which admits a nontrivial G-invariant partition B such that v = k + 1 ≥ 3 and

ΓB∼= Kmv+1. Moreover, each such graph Γ is isomorphic to Γ+(A; d, q), Γ=(A; d, q),

or Γ≃(A; d, q) (the third graph appears only when d ≥ 3). In each case we have

v = (qd − 1)/(q − 1) and the multiplicity m of D(B) for B ∈ B is equal to q − 1.

By a similar argument as in the proof of Theorem 9.4.2, one can prove the

following properties of the affine flag graphs above.

Theorem 9.5.2 Let d ≥ 2 and q ≥ 2 be a prime power, and set Θ := Θ(A; d, q).

Then the following (a)-(d) hold.

(a) Both Γ+(A; d, q) and Γ≃(A; d, q) are connected graphs with diameter two and

girth three, and with valencies (q−1)(qd−q) and (qd−q2)(qd−q)/(q−1), respectively.

(b) Γ=(A; d, q) has valency qd − q and contains (qd − 1)/(q − 1) connected com-

ponents, each of which is a complete qd−1-partite graph with q vertices in each part.

Moreover, Γ=(A; d, q) is an almost cover of Kqd.

(c) For distinct blocks Θ(σ),Θ(τ) of B(Θ), each vertex (σ, L) of Θ(σ) other than

(σ, Lστ ) is adjacent to exactly q − 1 vertices of Θ(τ) in Γ+(A; d, q), and adjacent

to exactly (qd − q2)/(q − 1) vertices of Θ(τ) in Γ≃(A; d, q). In particular, for Γ :=

Γ+(A; 2, q), Γ[Θ(σ),Θ(τ)] is isomorphic to Kq,q minus a perfect matching.


(d) For AGL(d, q) ≤ G ≤ AΓL(d, q), any G-symmetric graph with vertex set Θ

(under the induced action) is isomorphic to Γ+(A; d, q), or Γ≃(A; d, q), or Γ=(A; d, q),

or qd ·K(qd−1)/(q−1) with connected components the sets of flags incident with a com-

mon point, or qd−1(qd − 1)/(q − 1) ·Kq with connected components the sets of flags

incident with a common line.

Chapter 10

Local actions

To say that you know when you do know and say that you do not

know when you do not know – that is the way to acquire knowledge.


10.1 Introduction

In Section 3.2 we defined G(B) and G[B] to be the kernels of the actions of GB on B

and ΓB(B), respectively. In this chapter, we will study actions induced by these two

kernels. In particular, we will investigate the action of G[B] on B and the actions

of G(B) on ΓB(B), Γ(α) and ΓB(α) (where α ∈ B), and the influence of these “local

actions” on the structure of Γ. It is expected that the investigation in this chapter

would provide a basis for future study of imprimitive symmetric graphs. For our

purpose it seems natural to distinguish whether one of G(B), G[B] is a subgroup of

the other. With respect to this, we have the following (not necessarily exclusive)

possibilities: (i) G[B] ≤ G(B); (ii) G[B] 6≤ G(B); (iii) G(B) ≤ G[B]; (iv) G(B) 6≤ G[B];

(v) G[B] 6≤ G(B) and G(B) 6≤ G[B]. Setting M = G(B)G[B], then M GB and we have

Figure 6 in the lattice of subgroups of GB.

We will put our discussion in a general setting and consider the following sub-

groups of GB. Let d := diam(ΓB), which can be finite (if ΓB is connected) or ∞

(otherwise). For each integer i with 0 ≤ i < d+1, let ΓB(i, B) denote the set of blocks

of B with distance in ΓB no more than i from B. Then ΓB(i, B) is GB-invariant and

hence GB induces a natural action on ΓB(i, B). We will use G[i,B] to denote the

130 Local actions

kernel of this action, so in particular we have GB = G[0,B] and G[B] = G[1,B]. Figure

7 illustrates the relationships among these groups G[i,B] in the lattice of subgroups

of GB when d is finite.

s s

s

s

s

HH

HH

HH

HH

H

HH

HH

HH

HH

HG(B) G[B]

G(B) ∩G[B]

M = G(B)G[B]

GB

Figure 6 G(B) and G[B]

G[d,B]

G[d−1,B]

G[d−2,B]

G[3,B]

G[2,B]

G[B]

GB

s

s

sqqqs

s

s

sP

PP

PP

PP

s

s

s

q

q

q

s

s

s

G[d,B]G(B)

G[d−1,B]G(B)

G[d−2,B]G(B)

G[3,B]G(B)

G[2,B]G(B)

G[1,B]G(B)

s

s

s

q

q

q

s

s

s

G[d,B] ∩G(B)

G[d−1,B] ∩G(B)

G[d−2,B] ∩G(B)

G[3,B] ∩G(B)

G[2,B] ∩G(B)

G[1,B] ∩G(B)

hhhhhhhhhhhhhh

hhhhhhhhhhhhhh

hhhhhhhhhhhhhh

hhhhhhhhhhhhhh

hhhhhhhhhhhhhh

hhhhhhhhhhhhhh

sG(B)

hhhhhhhhhhhhhh

Figure 7 Relationships among G[i,B]’s

Introduction 131

The main results of this chapter are as follows. In Section 10.2, we will show

(Theorem 10.2.1) that each G[i,B] induces a G-invariant partition Bi of V (Γ) such

that the sequence B = B0,B1,B2, . . . ,Bi, . . . is a tower possessing some nice “level

structure” properties, where as in [66] a sequence of G-invariant partitions is said to

be a tower if each partition is a refinement of the previous partition. We will show

(Theorem 10.2.2) further that, if G[i,B] ≤ G(B) for some i ≥ 1 then G is faithful on

B; whilst if G[i,B] 6≤ G(B) for some i ≥ 1 then either Bi is a genuine refinement of B

or Γ is a multicover of ΓB. In Section 10.3 we will study a special case where, for

any C,D ∈ ΓB(B), either Γ(C) ∩ B = Γ(D) ∩ B, or Γ(C) ∩ Γ(D) ∩ B = ∅. Based

on the discussions in these two sections, we then study in Section 10.4 the case

where Γ is G-locally quasiprimitive. In this case we will show (Corollary 10.4.1)

amongst other things that, if B is a minimal G-invariant partition, then either

k = 1, or G[B] = G(B), or Γ is a multicover of ΓB. Recall that, for α ∈ V (Γ), we

use G[α] to denote the subgroup of Gα fixing setwise each block C ∈ ΓB(α). So G[α]

induces a natural action on Γ(α) ∩ C. In Section 10.4 we will also study G-locally

quasiprimitive graphs Γ such that G[α] is transitive on Γ(α) ∩ C, and prove that in

this case either Γ is a bipartite graph or Γ[B,C] is a matching.

As in most part of this thesis, we will identify in this chapter the blocks of D(B)

with the subsets Γ(C)∩B of B (with multiplicity m), for C ∈ ΓB(B). We conclude

this introductory section by making the following observations.

Lemma 10.1.1 Suppose Γ is a G-symmetric graph admitting a nontrivial G-invari-

ant partition B. Let B ∈ B and α ∈ B, and set d := diam(ΓB). Then the following

(a)-(f) hold.

(a) G(B) GB.

(b) G(B) Gα.

(c) G[α] Gα.

(d) G[i,B] GB for each integer i with 0 ≤ i < d + 1. In particular, we have

G[B] GB.

(e) G[i,B] G[i−1,B] for each integer i with 1 ≤ i < d+ 1.

(f) If i is at least the diameter of the connected components of ΓB, then G[i,B] is

equal to the kernel of the induced action of G on the component of ΓB containing

B. In particular, if ΓB is connected, then G[d,B] is equal to the kernel of the action

of G on B.

132 Local actions

Proof Since G(B) is the kernel of the action of GB on B, we have G(B) GB. Since

GB is transitive on B whilst G(B) is not, we have G(B) 6= GB and (a) is proved. From

this and G(B) ≤ Gα ≤ GB, we get (b) immediately. Similarly, (c) follows from the

fact that G[α] is the kernel of the action of Gα on ΓB(α). Since G[i,B] is the kernel

of the action of GB on ΓB(i, B), we have G[i,B] GB for each i with 0 ≤ i < d + 1.

In particular, we have G[B] = G[1,B] GB and thus (d) is proved. For 1 ≤ i < d+ 1,

since G[i,B] ≤ G[i−1,B] ≤ GB and since G[i,B] GB by (d), we get (e) immediately.

The G-symmetry of ΓB implies that its connected components are isomorphic and

hence have the same diameter. If i is no less than this diameter, then ΓB(i, B) is

equal to the set of blocks in the component of ΓB containing B and G induces an

action on ΓB(i, B). Hence the validity of the statements in (f) follows. 2

10.2 G-invariant partitions induced by G[i,B]

We first prove the following general result, which shows that each normal subgroup

of GB induces a refinement of the given G-invariant partition B.


ant partition B, and let B ∈ B. Then each normal subgroup N of GB induces a G-

invariant partition B∗N of V (Γ). Moreover, B∗

N is a refinement of B and the following

(a)-(c) hold.

(a) B∗N is the trivial partition α : α ∈ V (Γ) if and only if N ≤ G(B).

(b) B∗N coincides with B if and only if N is transitive on B.

(c) If N is a normal subgroup of G, then B∗N coincides with the G-normal parti-

tion BN of V (Γ) induced by N (defined after Lemma 2.2.2).

Proof Since NGB and GB is transitive on B, Lemma 2.2.2 implies that B∗ := αN

(for some α ∈ B) is a block of imprimitivity for GB in B. Since B is a G-invariant

partition of V (Γ), this implies that B∗ is a block of imprimitivity for G in V (Γ).

Hence B∗ induces a G-invariant partition of V (Γ), namely,

B∗N := (B∗)g : g ∈ G. (10.1)

The validity of (a)-(c) follows from the definition of B∗N immediately. 2

Partitions Induced by G[i,B] 133

Remark 10.2.1 For distinct blocks B,C ∈ B, there exists g ∈ G such that Bg = C.

So (GB)g = GC by Lemma 2.1.1(a), and hence N GB if and only if Ng GC . It

is easy to see that B∗Ng = B∗

N . So, in studying the G-invariant partition B∗N , we can

start from any chosen block B ∈ B.

In Lemma 10.1.1(d) we have seen that G[i,B] is a normal subgroup of GB, for

each integer i with 0 ≤ i < diam(ΓB) + 1. So it follows from Lemma 10.2.1 that

G[i,B] induces a G-invariant partition

Bi := Bgi : g ∈ G (10.2)

of V (Γ) which is a refinement of B, where Bi := αG[i,B] (for some α ∈ B) is a

typical block of Bi. Let vi, ri, bi, ki, si denote the parameters with respect to Bi,

as defined in Section 3.2. Since B0 is precisely the original partition B, we have

(v0, r0, b0, k0, s0) = (v, r, b, k, s). The following theorem gives some “level structure”

properties concerning these partitions.

Theorem 10.2.1 Suppose Γ is a G-symmetric graph admitting a nontrivial G-

invariant partition B. Let B ∈ B and set d := diam(ΓB). Then for each integer

i with 0 ≤ i < d + 1, G[i,B] induces a G-invariant partition Bi, defined in (10.2),

which is a refinement of B. Moreover, for 1 ≤ i < d+ 1, Bi is a refinement of Bi−1

and the following (a)-(d) hold.

(a) vi is a common divisor of vi−1 and ki−1, si is a divisor of si−1, and ri−1 is a

divisor of ri (with si−1/si = ri/ri−1).

(b) Each block of the 1-design D(Bi−1) (for Bi−1 ∈ Bi−1) is a disjoint union

of some blocks of Bi. More precisely, for adjacent blocks Bi−1, Ci−1 of ΓBi−1, G[i,B]

leaves Γ(Ci−1) ∩ Bi−1 invariant and the (G[i,B])-orbits on Γ(Ci−1) ∩ Bi−1 form a

(GBi−1,Ci−1)-invariant partition of Γ(Ci−1) ∩ Bi−1.

(c) ΓBi−1(α) = ΓBi−1

(β) for any vertices α, β in the same block of Bi.

(d) For each integer j with 0 ≤ j < i, the set Bi admits a G-invariant partition

Bij such that ΓBj∼= (ΓBi

)Bijand that the parameters vij, rij , bij, kij , sij with respect

to Bij satisfy vij = vj/vi, kij = kj/vi, bij = bj , rij = rj, sij = bi/rj.

Proof Let α ∈ B and Bi := αG[i,B] , and let Bi be as defined in (10.2) for each

i. Then, since G[i,B] GB by Lemma 10.1.1(d), Lemma 10.2.1 implies that Bi is

134 Local actions

a G-invariant partition of V (Γ) and is a refinement of B. For 1 ≤ i < d + 1,

since G[i,B] G[i−1,B] (Lemma 10.1.1(e)), it follows that Bi is a refinement of Bi−1.

Consequently, vi is a divisor of vi−1.

Now suppose Ci−1 is a block of Bi−1 adjacent to Bi−1 in ΓBi−1, and let C be

the block of B containing Ci−1. Then there exist β ∈ Γ(Ci−1) ∩ Bi−1 and γ ∈

Γ(Bi−1) ∩ Ci−1 such that β, γ are adjacent in Γ. By the definition of Bi−1, we have

Bi−1 = βG[i−1,B] and Ci−1 = γG[i−1,C] , and by Lemma 3.2.3(c) we have Γ(Ci−1) ∩

Bi−1 = βGBi−1,Ci−1 and Γ(Bi−1) ∩ Ci−1 = γGBi−1,Ci−1 . Note that B,C are adjacent

blocks of B. So we have ΓB(i − 1, C) ⊆ ΓB(i, B) and hence G[i,B] ≤ G[i−1,C]. This

implies that G[i,B] fixes Ci−1 setwise. Since G[i,B] G[i−1,B], G[i,B] also fixes Bi−1

setwise. Thus, we have G[i,B] ≤ GBi−1,Ci−1. This implies G[i,B] GBi−1,Ci−1

since

GBi−1,Ci−1≤ GB and G[i,B] GB (Lemma 10.1.1(d)). So G[i,B] leaves Γ(Ci−1)∩Bi−1

invariant and, again by Lemma 2.2.2, the (G[i,B])-orbits on Γ(Ci−1)∩Bi−1 constitute

a (GBi−1,Ci−1)-invariant partition of Γ(Ci−1)∩Bi−1. Thus, each block Γ(Ci−1)∩Bi−1

of the 1-design D(Bi−1) is a disjoint union of some blocks of Bi. This implies in

particular that vi is a divisor of ki−1, and so vi is a common divisor of vi−1 and ki−1.

One can see that each block Ci−1 of ΓBi−1(β) contains the same number of blocks

of ΓBi(β). Hence ri−1 is a divisor of ri. Since ri−1si−1 = risi = val(Γ), this implies

that si is a divisor of si−1.

If δ, ε are in the same block of Bi, without loss of generality we may suppose

that δ, ε ∈ Bi. Then since Bi is a (G[i,B])-orbit there exists x ∈ G[i,B] such that

δx = ε, and hence (ΓBi−1(δ))x = ΓBi−1

(ε). On the other hand, the elements of G[i,B]

fix setwise each block Ci−1 in ΓBi−1(Bi−1) since G[i,B] GBi−1,Ci−1

, as shown above.

In particular, x fixes setwise each block in ΓBi−1(δ) since ΓBi−1

(δ) ⊆ ΓBi−1(Bi−1).

Thus, we have ΓBi−1(δ) = (ΓBi−1

(δ))x = ΓBi−1(ε).

Let j be an integer with 0 ≤ j < i. Since for each ℓ with j + 1 ≤ ℓ ≤ i the

partition Bℓ is a refinement of the partition Bℓ−1, as shown above, we know that Bi

is a refinement of Bj and hence each block Cj of Bj is a union of some blocks of Bi.

Denote Cij = Bzi : Bz

i ⊆ Cj, z ∈ G, the set of blocks of Bi contained in Cj . Then

Bij := Cij : Cj ∈ Bj is a partition of Bi. We claim further that Bij is a G-invariant

partition of Bi under the induced action of G on Bi. In fact, if Cgij ∩ Cij 6= ∅ for

some g ∈ G, say (Bxi )g = By

i for some Bxi , B

yi ∈ Cij , then Bx

i , Byi ⊆ Cj and hence

(Bxi )g = By

i ⊆ Cj. Since Cj is a block of imprimitivity for G in V (Γ), this implies

Partitions Induced by G[i,B] 135

that g fixes Cj setwise. Therefore, we have Cgij = (Bz

i )g : Bz

i ⊆ Cj, z ∈ G = Cij

and hence Bij is G-invariant indeed. Clearly, the mapping ψ : Cj 7→ Cij is a

bijection from Bj to Bij. By the definition of a quotient graph, one can see that

ΓBj∼= (ΓBi

)Bijwith respect to ψ. Clearly, we have vij = vj/vi, kij = kj/vi, bij = bj

and rijsij = val(ΓBi) = bi. From vijrij = bijkij, we get (vj/vi)rij = bj(kj/vi), which

in turn implies rij = rj since vjrj = bjkj. Finally, we have sij = bi/rij = bi/rj and

the proof is complete. 2

Remark 10.2.2 If G[i,B] G for B ∈ B, then from Lemma 10.2.1(c), Bi is the

G-normal partition of V (Γ) induced by G[i,B]. In this case Γ is a multicover of ΓBi

(see [66, Section 1] or [71, Theorem 4.1]). This happens in particular for ΓB1 when

ΓB is a complete graph since in this case we have d = 1 and G[B] is the kernel of the

action of G on B.

Theorem 10.2.2 Suppose the triple (Γ, G,B) is as in Theorem 10.2.1. Let B ∈ B

and let an integer i satisfy 1 ≤ i < d + 1, where d := diam(ΓB). The one of the

following (a), (b) holds.

(a) G[i,B] ≤ G(B), in this case G is faithful on B if in addition G is faithful on

V (Γ).

(b) G[i,B] 6≤ G(B), and either

(i) G[i,B] induces the G-invariant partition Bi of V (Γ), defined in (10.2),

which is a genuine refinement of B such that vi is a common divisor of v

and k, si is a divisor of s, and r is a divisor of ri; or

(ii) Γ is a multicover of ΓB and G[i,B] is transitive on B.

Proof Suppose that G[i,B] ≤ G(B). Then, since G is transitive on B and since

G[i,Bg] = (G[i,B])g and G(Bg) = (G(B))

g for any g ∈ G, we have G[i,C] ≤ G(C) for all

blocks C ∈ B. Thus, if g is in the kernel of the action of G on B, then g ∈ G[i,C] in

particular and hence g ∈ G(C). In other words, g fixes each vertex in C. Since this

holds for all C, it follows that g fixes each vertex of Γ. So, if G is faithful on V (Γ),

then g = 1 and hence G is faithful on B as well.

136 Local actions

Now suppose that G[i,B] 6≤ G(B). Then, by Lemma 10.2.1(a), the partition Bi of

V (Γ) induced by G[i,B] is a nontrivial G-invariant partition of V (Γ). So we know

from Lemma 10.2.1(b) and Theorem 10.2.1 that, either Bi is a genuine refinement of

B, or G[i,B] is transitive on B. In the former case, it follows from Theorem 10.2.1(a)

that vi is a common divisor of v and k, si is a divisor of s and r is a divisor of ri,

and hence (i) in (b) occurs. Since G[i,B] fixes setwise the block B and each block

C ∈ ΓB(B), it also fixes setwise Γ(C) ∩ B. So in the latter case where G[i,B] is

transitive on B, we must have Γ(C) ∩ B = B, that is, Γ is a multicover of ΓB and

hence (ii) in (b) occurs. 2

Note that, if case (i) in Theorem 10.2.2(b) occurs, then at least one of the Bij

given in Theorem 10.2.1(d), say Bi0, is a nontrivial partition of Bi. If case (ii) in

Theorem 10.2.2(b) occurs, then from Lemma 10.2.1(b), the partition Bi induced by

G[i,B] coincides with B. Applying Theorem 10.2.2 to G[B], we have the following

consequence.

Corollary 10.2.1 Suppose the triple (Γ, G,B) is as in Theorem 10.2.1. Then one

of the following (a), (b) holds.

(a) G[B] ≤ G(B), in this case G is faithful on B if in addition G is faithful on

V (Γ).

(b) G[B] 6≤ G(B), and either

(i) G[B] induces a G-invariant partition of V (Γ), namely B1 defined in (10.2) for

i = 1, which is a genuine refinement of B such that v1 is a common divisor of v and

k, s1 is a divisor of s, and r is a divisor of r1; or

(ii) Γ is a multicover of ΓB and G[B] is transitive on B.

If the vertices in B are “distinguishable” in some sense, for example if ΓB(α) 6=

ΓB(β) for distinct α, β ∈ B, then case (a) in Corollary 10.2.1 occurs. (This happens

for G-symmetric graphs with k = v − 1 ≥ 1. See Theorems 4.2.1 and 4.3.1(c).)

If B is chosen to be a minimal nontrivial G-invariant partition of V (Γ), then case

(b)(i) in Corollary 10.2.1 does not appear. We conclude this section by giving the

following example which shows that case (b)(ii) in Corollary 10.2.1 occurs if G is

not quasiprimitive on V (Γ) and if B is a nontrivial G-normal partition of V (Γ).

Blocks of D(B) 137

Example 10.2.1 Suppose Γ is a G-symmetric graph, with G faithful but not

quasiprimitive on V (Γ). Then there exists a nontrivial normal subgroup N of G

which is intransitive on V (Γ), so the G-normal partition BN of V (Γ) induced by N

(see Lemma 2.2.2) is nontrivial. Let ΓN be the quotient graph of Γ with respect to

BN . Since N is contained in the kernel of the action of G on BN , G is not faithful on

BN . So from Corollary 10.2.1 we must have G[B] 6≤ G(B) for B ∈ BN . Since NG[B],

we have B = αN ⊆ αG[B] ⊆ B for α ∈ B, which implies αG[B] = B. Hence G[B] is

transitive on B, and consequently we come to the result (see e.g. [71, Theorem 4.1])

that Γ is a multicover of ΓN . Thus, case (b)(ii) in Corollary 10.2.1 occurs.

10.3 Two blocks of D(B) incident with either the

same or disjoint subsets of B

In Corollary 10.2.1 we have seen that, if G[B] 6≤ G(B), then either Γ is a multicover

of ΓB, or we get a genuine refinement of B. Note that GB is transitive on ΓB(B) and

G(B) GB by Lemma 10.1.1(a). So in the opposite case where G(B) 6≤ G[B], Lemma

2.2.2 implies that the G(B)-orbits on ΓB(B) form a nontrivial GB-invariant partition

of ΓB(B). Since G(B) fixes B pointwise, any two blocks in the same G(B)-orbit on

ΓB(B) induce repeated blocks of D(B). In some cases, blocks in distinct G(B)-orbits

on ΓB(B) may induce disjoint blocks of D(B). For example, in Remark 10.3.1 below

we will see that this happens in particular when Γ is G-locally quasiprimitive and

G(B) 6≤ G[B]. This motivates us to study the case where, for any C,D ∈ ΓB(B),

either Γ(C)∩B = Γ(D)∩B, or Γ(C)∩Γ(D)∩B = ∅. In this case, the multiplicity

m of D(B) is equal to r. This seemingly trivial case is by no means trivial because

it contains the following two very difficult but important subcases:

(i) k = 1;

(ii) k = v.

We have studied the first subcase (i) in Section 8.3, where we gave a construction

of such graphs from certain kinds of G-point- and G-block-transitive 1-designs. In

the second subcase (ii), Γ is a multicover of ΓB. Our study in this section shows

that (see Remark 10.3.2(a) below), in some sense, the study of G-symmetric graphs

with blocks Γ(C) ∩ B of D(B) (for C ∈ ΓB(B)) satisfying the condition above can

138 Local actions

be reduced to the study of these two subcases. The results obtained here will be

used in the next section.

Lemma 10.3.1 Suppose Γ is a G-symmetric graph admitting a nontrivial G-invar-

iant partition B. Let B ∈ B, α ∈ B, and let (a), (b), (c) be the following statements.

Then (a) implies (b), and (b) in turn implies (c).

(a) G(B) 6≤ G[B], and either Gα or (GB)ΓB(α) is quasiprimitive on ΓB(α);

(b) G(B) is transitive on ΓB(α);

(c) for C,D ∈ ΓB(B), either Γ(C) ∩ B = Γ(D) ∩B or Γ(C) ∩ Γ(D) ∩ B = ∅.

Proof (a) ⇒ (b) Suppose G(B) 6≤ G[B]. Then there exist x ∈ G(B) and C,D ∈

ΓB(B) with C 6= D such that Cx = D. Let α ∈ Γ(C)∩B, so that C ∈ ΓB(α). Since x

fixes each vertex inB and hence fixes α in particular, we have (Γ(α)∩C)x = Γ(α)∩D.

Since Γ(α) ∩ C 6= ∅, we have Γ(α) ∩ D 6= ∅ and hence D ∈ ΓB(α). Thus the

action of G(B) on ΓB(α) is nontrivial. On the other hand, since G(B) GB (Lemma

10.1.1(a)) and G(B) ≤ (GB)ΓB(α) ≤ GB, we have G(B) (GB)ΓB(α). So if (GB)ΓB(α)

is quasiprimitive on ΓB(α), then G(B) must be transitive on ΓB(α). Similarly, since

G(B) Gα (Lemma 10.1.1(b)) and G(B) acts on ΓB(α) in a nontrivial way, the

quasiprimitivity of Gα on ΓB(α) implies the transitivity of G(B) on ΓB(α).

(b) ⇒ (c) The assumption (b) and Lemma 3.2.6(b)(ii) together imply that G(B)

is transitive on ΓB(α) for each α ∈ B. That is, for any C,D ∈ ΓB(α), there exists

x ∈ G(B) such that Cx = D. This implies (Γ(C) ∩B)x = Γ(D) ∩B. However, since

x fixes each vertex in B, we have (Γ(C)∩B)x = Γ(C)∩B. So Γ(C)∩B = Γ(D)∩B.

In other words, if two blocks Γ(C)∩B,Γ(D)∩B of D(B) have a common vertex α,

then Γ(C) ∩ B = Γ(D) ∩B. Hence (c) is true. 2

Remark 10.3.1 Clearly, the quasiprimitivity of Gα on Γ(α) implies the quasiprim-

itivity of Gα on ΓB(α). So, if Γ is a G-locally quasiprimitive graph admitting a

nontrivial G-invariant partition B such that G(B) 6≤ G[B], then by Lemma 10.3.1,

either Γ(C) ∩ B = Γ(D) ∩B or Γ(C) ∩ Γ(D) ∩ B = ∅, for any C,D ∈ ΓB(B).

The main result in this section is the following theorem.

Theorem 10.3.1 Suppose Γ is a G-symmetric graph admitting a nontrivial G-

invariant partition B. Suppose further that, for any C,D ∈ ΓB(B), either Γ(C)∩B =

Blocks of D(B) 139

Γ(D)∩B or Γ(C)∩Γ(D)∩B = ∅. Then V (Γ) admits a second G-invariant partition

B∗ := (B∗)g : g ∈ G, where B∗ is a block of D(B). Moreover, the following (a)-(c)

hold.

(a) B∗ is a refinement of B, and it is a genuine refinement of B if and only if

2 ≤ k ≤ v − 1.

(b) Γ is a multicover of ΓB∗, k is a divisor of v, and the parameters v∗, r∗, b∗, k∗, s∗

with respect to B∗ satisfy v∗ = k∗ = k, b∗ = r∗ = r, s∗ = s.

(c) There exists a G-invariant partition B of B∗ such that (ΓB∗)B ∼= ΓB and the

parameters v, r, b, k, s with respect to B satisfy v = v/v∗, k = s = 1, b = b

and r = r.

Proof Our assumption on D(B) implies that the set of subsets of B of the form

Γ(C) ∩ B, for C ∈ ΓB(B), is a partition of B, which we denote by P(B). Thus

the blocks of P(B) have size k and k divides v. Let B∗ := Γ(C) ∩ B be a typical

block of P(B), where C ∈ ΓB(B). Since GB is transitive on ΓB(B) and since

(B∗)g = Γ(Cg) ∩B for g ∈ GB, we have P(B) = (B∗)g : g ∈ GB and hence P(B)

is a GB-invariant partition of B. We claim further that B∗ := (B∗)g : g ∈ G

defines a G-invariant partition of V (Γ). In fact, if (B∗)g ∩ B∗ 6= ∅ for some g ∈ G,

then Bg∩B 6= ∅ since B∗ ⊆ B and (B∗)g ⊆ Bg. But B is a block of imprimitivity for

G in V (Γ), so we have Bg = B and hence g ∈ GB. Thus (B∗)g ⊆ B and (B∗)g is a

block of P(B) having nonempty intersection with B∗. Since P(B) is a GB-invariant

partition of B, as shown above, this implies (B∗)g = B∗. Therefore, B∗ is a block of

imprimitivity for G in V (Γ) and so B∗ is a G-invariant partition of V (Γ). It is easily

checked that B∗ =⋃

B∈B P(B). Clearly, B∗ is a refinement of B, and it is a genuine

refinement of B if and only if 2 ≤ k ≤ v − 1. Since ΓB is G-symmetric, there exists

h ∈ G which interchanges B and C. So Γ(B) ∩ C = (Γ(C) ∩ B)h = (B∗)h ∈ B∗,

and hence each vertex in B∗ is adjacent to at least one vertex in (B∗)h. Therefore,

Γ is a multicover of ΓB∗ , and hence v∗ = k∗ = k, b∗ = r∗ = r, s∗ = s. Finally, it is

straightforward to show that B := P(B) : B ∈ B is a G-invariant partition of B∗

and that (ΓB∗)B ∼= ΓB. Also, it is clear that the parameters v, r, b, k, s with

respect to B are as specified in (c). 2

Remark 10.3.2 (a) The partition B∗ in Theorem 10.3.1 is equal to the trivial

partition α : α ∈ V (Γ) if and only if k = 1, and is equal to B if and only if

140 Local actions

k = v. In the general case where 2 ≤ k ≤ v − 1, B∗ is a genuine refinement of B,

and as k∗ = v∗, the partition (B∗)∗ resulting from applying Theorem 10.3.1 to B∗,

is equal to B∗. Moreover, the quotient graph ΓB∗ admits a G-invariant partition,

namely B, for which k = 1 and thus the construction given in Section 8.3 applies to

ΓB∗ .

(b) Setting i = 1 in Theorem 10.2.1(b), we know immediately that the partition

B1 (defined in (10.2) for i = 1) is a refinement of B∗. Moreover, B1 admits a G-

invariant partition B1 := P(B∗) : B∗ ∈ B∗, where P(B∗) := αG[B] ⊆ B∗ :

α ∈ B∗, such that (ΓB1)B1∼= ΓB∗ and ΓB1 is a multicover of ΓB∗ , and that the

parameters v1, r1, b1, k1, s1 with respect to B1 satisfy v1 = k1 = k/v1, r1 = b1 = r,

s1 = b1/r.

10.4 Locally quasiprimitive graphs

We now apply the results obtained in the last two sections toG-locally quasiprimitive

graphs. Such graphs were studied initially in [66, 67], and more recent results were

obtained in [54]. In this case we have the following theorem, which can be viewed

as a generalization of [43, Lemma 3.4].

Theorem 10.4.1 Suppose Γ is a G-locally quasiprimitive graph which admits a

nontrivial G-invariant partition B. Suppose further that G[B] 6= G(B).

(a) If G(B) 6≤ G[B], then G(B) is transitive on Γ(α) for each α ∈ B. Moreover,

either

(i) k = 1 and G[B] < G(B); or

(ii) k ≥ 2, k divides v, and V (Γ) admits a second nontrivial G-invariant

partition B∗ such that B∗ is a refinement of B, Γ is a multicover of ΓB∗

and the parameters v∗, r∗, b∗, k∗, s∗ with respect to B∗ satisfy v∗ = k∗ =

k, b∗ = r∗ = r, s∗ = s.

(b) If G[B] 6≤ G(B), then G[B] induces a nontrivial G-invariant partition B1 of V (Γ)

(defined in (10.2) for i = 1) such that B1 is a refinement of B, v1 is a common

divisor of v and k, s1 is a divisor of s, and r is a divisor of r1.

Locally Quasiprimitive Graphs 141

Proof (a) Suppose G(B) 6≤ G[B]. Then there exist x ∈ G(B) and distinct blocks

C,D of ΓB(B) such that Cx = D. Let α ∈ Γ(C)∩B, so Γ(α)∩C 6= ∅. Since x fixes

each vertex in B, it fixes α in particular and hence maps a vertex in Γ(α) ∩ C to a

vertex in Γ(α) ∩D. Since G(B) Gα (Lemma 10.1.1(b)), this implies that GΓ(α)(B) is

a nontrivial normal subgroup of GΓ(α)α . Therefore, by the G-local quasiprimitivity

of Γ, we conclude that G(B) is transitive on Γ(α). From Lemma 3.2.6(b)(i), this

assertion is true for all vertices α in B.

If k = 1, then ΓB(α) ∩ ΓB(β) = ∅ for distinct α, β ∈ B. Hence, if g ∈ GB

fixes each block C ∈ ΓB(B) setwise, then it also fixes each vertex in B. So we have

G[B] < G(B) in this case.

If k ≥ 2, then by Remark 10.3.1, for any C,D ∈ ΓB(B), either Γ(C) ∩ B =

Γ(D)∩B or Γ(C)∩Γ(D)∩B = ∅. Hence Theorem 10.3.1 applies, and the partition

B∗ defined therein is a nontrivial G-invariant partition of V (Γ) and is a refinement

of B. The truth of the remaining statements in (ii) follows from Theorem 10.3.1(b).

(b) Now we suppose G[B] 6≤ G(B). Then B1 := αG[B] has length at least two,

where α ∈ B. Hence it follows from Theorem 10.2.1 that the partition B1 (defined

in (10.2) for i = 1) is a nontrivial G-invariant partition of V (Γ) and is a refinement of

B, and that the parameters v1, s1, r1 with respect to B1 have the required properties.

2

For minimal nontrivial G-invariant partitions, we have the following consequence

of Theorem 10.4.1.

Corollary 10.4.1 Suppose Γ is a G-locally quasiprimitive graph, with G faithful on

V (Γ). Suppose further that B is a minimal nontrivial G-invariant partition of V (Γ).

Then one of the following (a)-(c) holds.

(a) G[B] = G(B), in this case G is faithful on B;

(b) G[B] < G(B) and k = 1;

(c) Γ is a multicover of ΓB.

Moreover, if ΓB is a complete graph, then the occurrence of (a) implies G[B] = G(B) =

1; if G[B] 6≤ G(B), then the occurrence of (c) implies that G[B] is transitive on B.

Proof In the case where G(B) = G[B], G is faithful on B by Corollary 10.2.1(a).

Suppose G(B) 6= G[B]. Then either G(B) 6≤ G[B] or G[B] 6≤ G(B). In the former case,

142 Local actions

Theorem 10.4.1(a) applies. If (i) in Theorem 10.4.1(a) appears, then we have k = 1

and G[B] < G(B), and hence (b) above occurs. If (ii) in Theorem 10.4.1(a) appears,

then by the minimality of B, the partition B∗ therein must coincide with B; hence

Γ is a multicover of ΓB and (c) holds. In the latter case where G[B] 6≤ G(B), by

Corollary 10.2.1 and the minimality of B, we know that Γ is a multicover of ΓB

(hence (c) above occurs), and moreover G[B] is transitive on B.

Now suppose that ΓB is a complete graph, and that case (a) occurs. Then G[B]

is the kernel of the action of G on B and hence G[B] = G(B) G. This implies

that G(B) = g−1G(B)g = G(Bg) for any g ∈ G. Since Bg runs over all blocks of B

when g runs over G, this means that G(B) fixes each vertex of Γ, and hence by the

faithfulness of G on V (Γ) we get G[B] = G(B) = 1. 2

Recall that G[α] is the subgroup of Gα fixing setwise each block in ΓB(α). So G[α]

induces an action on Γ(α)∩C, for each C ∈ ΓB(α). As exemplified in the following

lemma, it may happen that G[α] is transitive on Γ(α)∩C, or, equivalently, Γ(α)∩C

is a (G[α])-orbit on Γ(α).


ant partition B, and let α ∈ V (Γ). If Gα is regular on ΓB(α), then G[α] is transitive

on Γ(α) ∩ C, for each C ∈ ΓB(α).

Proof For any C ∈ ΓB(α) and β, γ ∈ Γ(α)∩C, by the G-symmetry of Γ there exists

x ∈ Gα such that βx = γ, and hence x fixes C setwise. Since by our assumption

Gα acts regularly on ΓB(α), this implies that Dx = D for all D ∈ ΓB(α), and hence

x ∈ G[α]. Thus, any vertex β in Γ(α) ∩ C can be mapped to any other vertex γ in

Γ(α) ∩ C by an element of G[α]. In other words, G[α] is transitive on Γ(α) ∩ C. 2

We conclude this section by studying G-locally quasiprimitive graphs Γ such that

G[α] is transitive on Γ(α) ∩ C, for C ∈ ΓB(α). In this case we have the following

theorem which is a counterpart of Corollary 3.2.1.

Theorem 10.4.2 Suppose Γ is a G-locally quasiprimitive graph admitting a non-

trivial G-invariant partition B. Suppose further that G[α] is transitive on Γ(α)∩B,

for some α ∈ V (Γ) and B ∈ ΓB(α). Then either

(a) Γ[B,C] ∼= k ·K2 is a matching of k edges, for adjacent blocks B,C of B; or

Locally Quasiprimitive Graphs 143

(b) Γ is a bipartite graph with each part of the bipartition of a connected compo-

nent contained in some block of B, and for any C,D ∈ ΓB(B), either Γ(C) ∩ B =

Γ(D) ∩B or Γ(C) ∩ Γ(D) ∩ B = ∅ (hence v = bk).

Proof From Lemma 3.2.6(c)(ii), our assumption on G[α] implies that G[α] is transi-

tive on Γ(α)∩C for each C ∈ ΓB(α). So, if GΓ(α)[α] = 1, then we have |Γ(α)∩C| = 1.

That is, Γ[B,C] is a matching for adjacent blocks B,C of B, and hence (a) holds.

In the following we suppose that GΓ(α)[α] 6= 1. Then, since G

Γ(α)[α] GΓ(α)

α by Lemma

10.1.1(c) and since Γ is G-locally quasiprimitive by our assumption, G[α] must be

transitive on Γ(α). However, G[α] fixes Γ(α)∩C setwise for each C ∈ ΓB(α). So we

must have r = |ΓB(α)| = 1 and hence Γ(α) ⊆ C for some C. Let B be the block

of B containing α. Then, since G is transitive on arcs of Γ, for any β ∈ Γ(α) there

exists an element of G which interchanges α and β and hence interchanges B and

C. Hence Γ(α) ⊆ C implies Γ(β) ⊆ B. Similarly, Γ(β) ⊆ B implies Γ(γ) ⊆ C

for any γ ∈ Γ(β). Continuing this process, one can see that Γ[B,C] consists of

connected components of Γ, and hence each such component is a bipartite graph

with the two parts of the bipartition contained in B,C, respectively. This, together

with the fact r = 1, implies that Γ is a bipartite graph with v = bk, and that either

Γ(C) ∩ B = Γ(D) ∩B or Γ(C) ∩ Γ(D) ∩ B = ∅ for any C,D ∈ ΓB(B). 2

From Lemma 10.4.1, the results in Theorem 10.4.2 hold in particular when Γ is

a G-locally quasiprimitive graph such that Gα is regular on ΓB(α) for α ∈ V (Γ). In

this case, if Γ is not a bipartite graph, then Γ[B,C] is a matching and hence, by

Lemma 3.2.4(b), Gα is regular on Γ(α). Hence G is regular on the arcs of Γ if in

addition Γ is connected. Examples of such graphs include G-Frobenius graphs [32,

Definition 1.2] arising from self-paired G-orbitals of a Frobenius group G.

144 Local actions

Chapter 11

Local actions: Heritage of thelabelling method

He who by reanimating the Old can gain knowledge of the New is

qualified to teach others.


Continuing our study on “local actions”, we will investigate in this chapter the

particular case where the induced actions ofGB on B and ΓB(B) are permutationally

equivalent. That is, we will study G-symmetric graphs Γ admitting a nontrivial G-

invariant partition B such that the following [P] holds for some, and hence all (see

Lemma 3.2.6(a)), blocks B of B.

[P] The induced actions of GB on B and ΓB(B) are permutationally equivalent

with respect to some bijection ρ : B → ΓB(B).

From a geometric point of view, this requires that the automorphism group of D(B)

induced by GB (Lemma 3.2.5) acts in essentially the same way on the points and the

blocks of the 1-design D(B) = (B,ΓB(B), I). Clearly, any G-symmetric graph such

that k = v − 1 ≥ 2 and D(B) contains no repeated blocks possesses this property

(see Example 11.1.1 below), and this observation is one of the motivations for the

study in this chapter. Recall that in this case ΓB is (G, 2)-arc transitive (Theorem

5.1.2); we will characterize such graphs as the only graphs Γ satisfying [P] such

that ΓB is (G, 2)-arc transitive (Theorems 11.3.1(c)). Under the assumption [P],

we will develop a labelling technique similar to that used in Section 5.1, and we

146 Heritage of labelling method

will show that D(B) plays a more active role in influencing Γ, ΓB and Γ[B,C]. We

will study the case where in addition the bijection ρ in [P] preserves the incidence

relation of D(B) in the sense that, for α ∈ B and C ∈ ΓB(B), αIC if and only if

ρ−1(C)Iρ(α). Finally, based on the labelling technique, we will prove that the class

of G-symmetric graphs satisfying [P] is precisely the class of 3-arc graphs Ξ(Σ,∆) of

G-symmetric graphs Σ with respect to self-paired G-orbits ∆ on Arc3(Σ). Therefore,

this chapter may be viewed as an extension of Chapter 5. To avoid triviality, we

assume val(Γ) > 1 in this chapter.

11.1 Examples

Example 11.1.1 Let Γ be a G-symmetric graph admitting a nontrivial G-invariant

partition B such that k = v − 1 ≥ 2 and D(B) contains no repeated blocks. Then,

for each α ∈ B, the set B(α) defined in (4.1) contains a unique block Cα and, by

Theorem 4.3.2(a), [P] is satisfied for the bijection ρ : α 7→ Cα from B to ΓB(B).

Graphs in Example 11.1.1 have been studied in Chapters 5-7. The following

example shows that, besides these graphs, there exist other G-symmetric graphs

for which [P] is satisfied. Note that in this example we have k < v − 1 and D(B)

contains repeated blocks.

Example 11.1.2 Let PG(2, 2) be the Fano plane whose points 1, 2, . . . , 7 are as

shown in Figure 8. Let X be the set of ordered pairs of distinct points of PG(2, 2).

Then G := PGL(3, 2) is transitive on X ([10, Theorem 2.5.4]). Define Γ to be the

graph with vertex set X in which αβ, γδ ∈ X are adjacent if and only if (i) α, β, γ, δ

are distinct, and (ii) β, δ and the unique point collinear with α, γ are distinct and

are collinear in PG(2, 2). For example, 17, 26 are adjacent in Γ since the unique

point collinear with 1, 2 is 3 and since 7, 6, 3 are collinear in PG(2, 2). Similarly,

one can see that Γ(17) = 26, 62, 35, 53. Note that the pointwise stabilizer G17

of 1, 7 in G contains an element which exchanges 2 and 6, and exchanges 3 and

5; also G17 contains an element which exchanges 2 and 3, and exchanges 6 and

5. So G17 is transitive on Γ(17), and hence Γ is G-symmetric. One can see that

Γ ∼= 7 ·K2,2,2 and B := B(σ) : σ is a point of PG(2, 2) is a G-invariant partition

of X, where B(σ) := στ : τ is a point of PG(2, 2) with τ 6= σ. We have ΓB∼= K7,

Labelling Technique 147

Γ[B(σ), B(τ)] ∼= 4 ·K2 for σ 6= τ , D(B(1)) is a 1-(6, 4, 4) design, and the traces of

the blocks of D(B(1)) are 12, 13, 14, 17, 14, 15, 16, 17, 12, 13, 16, 15 with each

repeated twice. Thus the block size of D(B(1)) is less than |B(1)| − 1. Clearly, the

induced actions of GB(σ) on B(σ) and ΓB(B(σ)) are permutationally equivalent with

respect to the bijection ρ : στ 7→ B(τ). Note that στ is adjacent to a vertex in a

block B(δ) if and only if σδ is adjacent to a vertex in the block B(τ).

1

3

2 67

5 4

Figure 8 Fano plane

11.2 The labelling technique

As a fundamental fact, we now show that [P] holds if and only if the vertices of

Γ can be labelled in a natural way by the arcs of ΓB. For convenience, we call a

mapping µ : V (Γ) → Arc(ΓB) compatible with B if, for any α ∈ V (Γ), the arc µ(α)

of ΓB is initiated at the block B(α) containing α.


invariant partition B. Then [P] holds if and only if the actions of G on V (Γ) and

Arc(ΓB) are permutationally equivalent with respect to some bijection µ : V (Γ) →

Arc(ΓB) compatible with B. Moreover, in this case we have b = v ≥ 2, G[B] = G(B),

and G is faithful on B if G is faithful on V (Γ).

Proof Suppose first that [P] holds for some B ∈ B and a bijection ρ : B → ΓB(B),

and let α be a fixed vertex of B. Then, since Γ is G-vertex-transitive, each vertex


of Γ has the form αx for some x ∈ G. We will show that µ : αx 7→ (Bx, (ρ(α))x),

x ∈ G, defines a bijection from V (Γ) to Arc(ΓB) which is compatible with B. In

fact, if αx = αy for some x, y ∈ G, then xy−1 ∈ Gα (≤ GB), and hence Bxy−1=

B and (ρ(α))xy−1

= ρ(αxy−1

) = ρ(α). Therefore, we have µ(αx) = µ(αy) and

thus µ is well-defined. Secondly, if µ(αx) = µ(αy) for two vertices αx, αy, then

xy−1 ∈ GB since Bx = By. This, together with (ρ(α))x = (ρ(α))y, implies that

ρ(α) = (ρ(α))xy−1

= ρ(αxy−1

). Note that xy−1 ∈ GB implies αxy−1

∈ B, and that ρ

is a bijection from B to ΓB(B). So we have αxy−1

= α, implying αx = αy and hence

µ is injective. Since G is transitive on arcs of ΓB, µ is in fact a bijection from V (Γ) to

Arc(ΓB). Since B and ρ(α) are adjacent blocks and Bx = (B(α))x = B(αx), Bx and

(ρ(α))x are adjacent blocks and hence µ is compatible with B. It follows from the

definition that the actions of G on V (Γ) and Arc(ΓB) are permutationally equivalent

with respect to µ. Moreover, the definition of µ does not depend on the choice of

α ∈ B. In fact, for another vertex β ∈ B and any vertex of Γ, say γ = αx = βy

for some x, y ∈ G, we have Bx = B(αx) = B(βy) = By and hence xy−1 ∈ GB.

So (ρ(α))xy−1

= ρ(αxy−1

) = ρ(β), implying (B, ρ(α))x = (B, ρ(β))y and indeed the

definition of µ is independent of the choice of α ∈ B.

Now suppose conversely that the actions of G on V (Γ) and Arc(ΓB) are permuta-

tionally equivalent with respect to a bijection µ : V (Γ) → Arc(ΓB) which is compat-

ible with B. Then (B, ρ(α)) = µ(α), for α ∈ B, defines a bijection ρ : B → ΓB(B).

It is easily checked that the actions of GB on B and ΓB(B) are permutationally

equivalent with respect to ρ.

Finally, if [P] holds, then b = |ΓB(B)| = |B| = v ≥ 2 and G[B] = G(B) for each

B ∈ B. So if G is faithful on V (Γ) then it is faithful on B as well. 2

Lemma 11.2.1 implies that, under the assumption [P], each vertex α of Γ can

be uniquely labelled by an ordered pair “BC” of adjacent blocks of ΓB, where

(B,C) = µ(α). In the following we will identify α with the label “BC”, so we have

G“BC” = GB,C . Since (µ(α))x = µ(αx), it follows that

“BC”x = “BxCx” (11.1)

for x ∈ G and “BC” ∈ V (Γ). One can see that the block B is precisely the set of

those vertices of Γ whose labels have the first coordinate B, that is, B = “BC” :

(B,C) ∈ Arc(ΓB). Note that each vertex α = “BC” of Γ has a unique mate


α′ := “CB”, and that z : α 7→ α′ defines an involution on V (Γ). Also, z centralises

G since “BC”zx = “CB”x = “CxBx” = “BxCx”z = “BC”xz for any x ∈ G. Since

G preserves B invariant whilst it is easy to see that Bz = α′ : α ∈ B 6∈ B, we

have z 6∈ G. Clearly, α, α′ : α ∈ V (Γ) is a (G × 〈z〉)-invariant partition of

V (Γ), and the graph Γ′ with vertex set V (Γ) and arc set (α, α′) : α ∈ V (Γ) is

G-symmetric. We record these basic results in the following theorem, which will be

used repeatedly in our later discussion. The validity of these results for the graphs

in Example 11.1.1 has been established in Section 5.1.

Theorem 11.2.1 Suppose that Γ is a G-symmetric graph admitting a nontrivial

G-invariant partition B such that the actions of GB on B and ΓB(B) are permu-

tationally equivalent, for some B ∈ B. Let µ : V (Γ) → Arc(ΓB) be the bijection

guaranteed by Lemma 11.2.1. Then the following (a)-(d) hold.

(a) Each vertex α of Γ can be labelled uniquely by an ordered pair “BC” of

adjacent blocks of ΓB, where (B,C) = µ(α). Moreover, we have G“BC” = GB,C and

“BC”x = “BxCx” for “BC” ∈ V (Γ) and x ∈ G.

(b) Each vertex α = “BC” has a unique mate α′ := “CB”, the mapping z : α 7→

α′ defines an involution such that z 6∈ G and z centralises G, P := α, α′ : α ∈

V (Γ) is a (G × 〈z〉)-invariant partition of V (Γ), and the graph Γ′ with vertex set

V (Γ) and arc set (α, α′) : α ∈ V (Γ) is G-symmetric.

(c) B∗ := B∗ : B ∈ B (where B∗ := Bz) is a G-invariant partition of V (Γ),

GB = GB∗, and the actions of GB on B and B∗ are transitive and permutationally

equivalent with respect to the restriction of z on B.

(d) There is no edge of Γ joining vertices of B and B∗. In particular, for each

arc (“BC”, “DE”) of Γ, (C,B,D,E) is a 3-arc of ΓB.

Proof The truth of (a) and (b) has been shown above, and from this we get (c) by

a routine argument. To prove (d), we assume B,C are two adjacent blocks of ΓB.

If “CB” is adjacent to “BC”, then, since val(Γ) > 1, “CB” is adjacent to a vertex

“B1C1” distinct from “BC”. By the G-symmetry of Γ, there exists x ∈ G such that

(“CB”, “BC”)x = (“CB”, “B1C1”), which implies C = Cx = C1, B = Bx = B1.

This is a contradiction and hence each vertex “CB” of V (Γ) is not adjacent to

its mate “BC”. Similarly, if “CB” is adjacent to a vertex “BD” ∈ B \ “BC”,

then we can take a vertex “B1D1” which is distinct from “BD” and is adjacent


to “CB”. By the G-symmetry of Γ, we have (“CB”, “BD”)x = (“CB”, “B1D1”)

for some x ∈ G, and hence B = Bx = B1. On the other hand, there exists

y ∈ G such that (“CB”, “BD”)y = (“B1D1”, “CB”). This implies C = By = D1,

and hence “B1D1” = “BC”. Again, this is a contradiction and hence there is no

edge of Γ between B and B∗. In particular, if (“BC”, “DE”) is an arc of Γ, then

C 6= D,B 6= E and hence (C,B,D,E) is a 3-arc of ΓB. 2

The following theorem is a counterpart of Theorem 5.1.2(a)(b).


G-invariant partition B such that the actions of GB on B and ΓB(B) are permuta-

tionally equivalent, for some B ∈ B. Then one, and only one, of the following (a),

(b) occurs.

(a) Any two adjacent vertices have labels involving four distinct blocks. In this

case, each block of B∗ is an independent set of Γ.

(b) Any two adjacent vertices of Γ share the same second coordinate. In this case,

Γ is disconnected with each block of B∗ consisting of connected components of Γ,

and moreover we have girth(ΓB) = 3, Γ[B,C] ∼= k ·K2 and val(Γ) = |DGB,C |, where

B,C,D ∈ B such that “CB”, “DB” are adjacent in Γ. In particular, Γ[B∗] ∼= Kv if

and only if ΓB is (G, 2)-arc transitive, and in this case we have Γ ∼= n(v + 1) ·Kv,

Γ[B,C] ∼= (v − 1) · K2 and ΓB∼= n ·Kv+1 for an integer n, and the group induced

on the vertex set of a connected component of ΓB is 3-transitive.

Proof If there exist two adjacent vertices of Γ, say “CB”, “DB”, which share

the same second coordinate. Then, since Γ is G-symmetric, by Theorem 11.2.1(a)

any arc of Γ has the form (“CxBx”, “DxBx”), for some x ∈ G, and hence any

two adjacent vertices of Γ share the same second coordinate. Thus, either (a) or

(b) occurs. It is easy to see that (a) occurs if and only if each block of B∗ is an

independent set of Γ.

In the following we suppose that (b) occurs, and let “CB”, “DB” be adjacent

vertices. Then any two adjacent vertices of Γ lie in the same block of B∗. Hence

the subgraph Γ[E∗] induced by each E∗ ∈ B∗ consists of connected components

of Γ. Clearly, we have girth(ΓB) = 3 since (B,C,D,B) is a triangle of ΓB. By

our assumption, “CB” is the unique vertex in C adjacent to “DB”. So we have


Γ[C,D] ∼= k ·K2. Moreover, a vertex “D1B” ∈ B∗ is adjacent to “CB” in Γ ⇔ there

exists g ∈ G such that (“CB”, “DB”)g = (“CB”, “D1B”) ⇔ there exists g ∈ GB,C

such that Dg = D1. Thus, we have val(Γ) = |DGB,C |. In particular, Γ[B∗] ∼= Kv ⇔

GB,C is transitive on ΓB(B) \ C ⇔ GB is 2-transitive on ΓB(B) ⇔ ΓB is (G, 2)-

arc transitive. In this case, the argument above shows that (i) Γ ∼= |B∗| · Kv, (ii)

B ∪ ΓB(B) induces the complete graph Kv+1 which is a connected component of

ΓB (note that b = v by Lemma 11.2.1), and (iii) G induces a 3-transitive group

on B ∪ ΓB(B). Therefore, we have ΓB∼= n · Kv+1 and Γ ∼= n(v + 1) ·Kv for an

integer n. Counting the number of edges of Γ in two ways, we get (n(v+ 1)v/2)k =

n(v + 1)(v(v− 1)/2), which implies k = v − 1 and hence Γ[C,D] ∼= (v − 1) ·K2. 2

Note that case (a) in Theorem 11.2.2 occurs when girth(ΓB) ≥ 4. If girth(ΓB) ≥

5, then we get the following generalizations of Theorem 5.1.3 and Corollary 5.1.2 –

the proofs are very much similar and hence ommitted.



tionally equivalent, for some B ∈ B. Suppose further that girth(ΓB) ≥ 5. Then the

following (a)-(c) hold.

(a) Γ[α, α′, β, β ′] ∼= K2 for adjacent blocks α, α′ and β, β ′ of P.

(b) Γ[B∗, C∗] is a matching for adjacent blocks B∗, C∗ of B∗; in particular we

have Γ[B∗, C∗] ∼= K2 if girth(ΓB) ≥ 7.

(c) The involution z : α 7→ α′ (α ∈ V (Γ)) defines a graph monomorphism from

Γ to the complement Γ. Moreover, z induces graph monomorphisms from ΓB to ΓB∗,

and from ΓB∗ to ΓB, defined by B 7→ B∗, and B∗ 7→ B, respectively.

Corollary 11.2.1 With the same assumptions as in Theorem 11.2.3, we have val(Γ)

≤ (|V (Γ)| − 2)/4 and val(ΓB∗) ≤ (|V (Γ)|/v) − v − 1. If in addition girth(ΓB) ≥ 7,

then val(Γ) ≤ (|V (Γ)|/v2) − (1/v) − 1.

Remark 11.2.1 Let k∗ denote the block size of the 1-design D(B∗). If k∗ = 1,

then val(ΓB∗) = v · val(Γ) > v = |B∗| and hence the actions of GB∗ on B∗ and

ΓB∗(B∗) cannot be permutationally equivalent. From Theorem 11.2.3(b), this is the

case in particular when girth(ΓB) ≥ 7. Thus the G-invariant partition B∗ may not


satisfy [P]. Moreover, in the case where k∗ = 1, the construction given in Section 8.3

applies, and so Γ is isomorphic to a certain G-flag graph of the 1-design D(Γ,B∗).

11.3 The 1-design D(B)

Part (d) of Theorem 11.2.1 is equivalent to saying that, if (“BC”, D) is a flag of

D(B), then C 6= D and hence (C,B,D) is a 2-arc of ΓB. Denote by arc2(ΓB) the

set of all such 2-arcs of ΓB, that is,

arc2(ΓB) := (C,B,D) : “BC”ID.

As before, denote by D∗(B) the dual 1-design of D(B). Then we have the following

theorem which conveys more information about the 1-design D(B).



tionally equivalent, for some B ∈ B. Then the following (a)-(d) hold.

(a) Both D(B) and D∗(B) are 1-(v, k, k) designs.

(b) arc2(ΓB) is a G-orbit on Arc2(ΓB), k = |CGB,D |, and k + m ≤ v, where

(C,B,D) ∈ arc2(ΓB) and m is the multiplicity of D(B).

(c) The following conditions (i)-(iv) are equivalent:

(i) ΓB is (G, 2)-arc transitive;

(ii) arc2(ΓB) = Arc2(ΓB);

(iii) k = v − 1;

(iv) k = v − 1 and D(B) contains no repeated blocks.

(d) Γ[B,C] ∼= Kk,k if and only if GB,C,D is transitive on Γ(B)∩D for (C,B,D) ∈

arc2(ΓB). In particular, Γ[B,C] ∼= Kv−1,v−1 if and only if ΓB is (G, 3)-arc transitive.

Proof (a) That D(B) is a 1-design implies vr = bk. Since b = v (Lemma 11.2.1),

we have r = k and hence both D(B) and D∗(B) are 1-(v, k, k) designs.

(b) Let (C,B,D), (C1, B1, D1) ∈ arc2(ΓB). Then “BC” is adjacent to a vertex

β ∈ D and “B1C1” is adjacent to a vertex β1 ∈ D1. So “BxCx” is adjacent to

βx ∈ Dx for any x ∈ G. Thus (Cx, Bx, Dx) ∈ arc2(ΓB) and hence arc2(ΓB) is G-

invariant. On the other hand, since Γ is G-symmetric, there exists y ∈ G such that

(“BC”, β)y = (“B1C1”, β1), which implies (C,B,D)x = (C1, B1, D1) and hence G

The 1-Design D(B) 153

is transitive on arc2(ΓB). Therefore, arc2(ΓB) is a G-orbit on Arc2(ΓB). From this

we have: “BE” ∈ B is adjacent to a vertex in D ⇔ (E,B,D) ∈ arc2(ΓB) ⇔ there

exists x ∈ G such that (C,B,D)x = (E,B,D) ⇔ there exists x ∈ GB,D such that

Cx = E. So we have k = |CGB,D |. Now suppose D1, . . . , Dm ∈ ΓB(B) are repeated

blocks of D(B), so we have Γ(D1) ∩ B = · · · = Γ(Dm) ∩ B. Then by Theorem

11.2.1(d), none of the m distinct vertices “BD1”, . . . , “BDm” of B is in Γ(D1) ∩B,

and hence k +m ≤ v follows.

(c) Clearly, (i) and (ii) are equivalent since arc2(ΓB) is a G-orbit on Arc2(ΓB).

Note that k = v − 1 implies k = v − 1 ≥ 2 for otherwise we would have val(Γ) = 1,

contradicting our assumption on the valency of Γ. From the argument in the proof

of (b), we have: k = v − 1 ⇔ k = v − 1 ≥ 2 ⇔ GB,D is transitive on ΓB(B) \ D

⇔ GB is 2-transitive on ΓB(B) ⇔ ΓB is (G, 2)-arc transitive. So (i) and (iii) are

equivalent. Clearly, (iv) implies (iii). Conversely, since k+m ≤ v as we have shown

above, k = v − 1 implies m = 1 and hence D(B) has no repeated blocks. The

equivalence of (i)-(iv) is then established.

(d) Let (C,B,D) ∈ arc2(ΓB). Then by the G-symmetry of Γ, G“BC”,D = GB,C,D

is transitive on Γ(“BC”) ∩D 6= ∅. Clearly, we have: Γ[B,D] ∼= Kk,k ⇔ Γ(“BC”) ∩

D = Γ(B) ∩ D ⇔ G“BC”,D is transitive on Γ(B) ∩ D ⇔ GB,C,D is transitive on

Γ(B)∩D. In particular, from (c) above and Theorem 11.2.1(d) we have: Γ[B,D] ∼=

Kv−1,v−1 ⇔ k = v − 1 and GB,C,D is transitive on D \ “DB” ⇔ ΓB is (G, 2)-arc

transitive and GB,C,D is transitive on ΓB(D) \ B ⇔ ΓB is (G, 3)-arc transitive. 2

Remark 11.3.1 (a) Applying Theorem 11.3.1(c) to the graphs Γ in Example 11.1.1,

we recover the result (Theorem 5.1.2) that, if k = v − 1 ≥ 2 and D(B) contains no

repeated blocks, then ΓB is (G, 2)-arc transitive. Furthermore, Theorem 11.3.1(c)

shows that, under the assumption [P], this is the only case where ΓB is (G, 2)-arc

transitive. Part (d) of Theorem 11.3.1 implies that in such a case ΓB is (G, 3)-arc

transitive if and only if Γ[B,C] ∼= Kv−1,v−1 (Theorem 5.3.1), and that this is the

only case where ΓB is (G, 3)-arc transitive.

(b) In Theorem 11.2.2(b) we have proved that, if adjacent vertices of Γ share the

same second coordinate, then Γ[B∗] ∼= Kv if and only if ΓB is (G, 2)-arc transitive.

By Theorem 11.3.1(c), this in turn is true if and only if k = v − 1 ≥ 2 and D(B)

contains no repeated blocks. So the assertions in the last sentence of Theorem


11.2.2(b) concerning Γ, Γ[B,C], ΓB and the group induced on a component of ΓB

can be derived from Theorem 5.1.2(b).

11.4 The case where ρ is incidence-preserving

In this section, we study the case where the bijection ρ in [P] is incidence-preserving,

that is, it satisfies

αID ⇔ ρ−1(D)Iρ(α) (11.2)

for α ∈ B and D ∈ ΓB(B). Using labels for the vertices of Γ, this condition can be

restated as

“BC”ID ⇔ “BD”IC (11.3)

for distinct C,D ∈ ΓB(B), which in turn is equivalent to saying that

(C,B,D) ∈ arc2(ΓB) ⇔ (D,B,C) ∈ arc2(ΓB). (11.4)

Thus, in view of Theorem 11.3.1(b), one of the above holds if and only if arc2(ΓB) is

a self-paired G-orbit on Arc2(ΓB). By Theorem 11.3.1(c) this is the case in particular

when Γ is as in Example 11.1.1. However, there are other cases for which (11.3) is

satisfied. This happens for the graph Γ in Example 11.1.2, where (11.3) is satisfied

(see the last sentence in that example) but ΓB is not (G, 2)-arc transitive by Theorem

11.3.1(c) and the fact that 4 = k < v − 1 = 5.

The additional requirement above implies immediately that D(B) is a self-dual

1-design, as stated below.

Proposition 11.4.1 Suppose that Γ is a G-symmetric graph admitting a nontriv-

ial G-invariant partition B such that, for some B ∈ B, the actions of GB on B

and ΓB(B) are permutationally equivalent with respect to an incidence-preserving

bijection ρ. Then D(B) is a self-dual 1-(v, k, k) design and ρ induces a polarity of

D(B).

Proof Let ψ be the bijection from B∪ΓB(B) to ΓB(B)∪B defined by ψ(α) = ρ(α),

ψ(C) = ρ−1(C) for α ∈ B and C ∈ ΓB(B). Then ψ(B) = ΓB(B), ψ(ΓB(B)) = B,

and (11.2) implies that αIC ⇔ ψ(C)Iψ(α) ⇔ ψ(α)I∗ψ(C). Thus, ψ is an isomor-

phism from D(B) to D∗(B) and hence D(B) is self-dual. Clearly, we have ψ2 = 1

and hence ψ is a polarity of D(B). 2

Incidence-preserving Case 155

For brevity we call a chordless 6-cycle in a given graph a hexagon. Recall that

in Section 11.2 we defined Γ′ to be the graph with vertex set V (Γ) and edge set

α, α′ : α ∈ V (Γ). In the case where (b) in Theorem 11.2.2 occurs, we have the

following result which is interesting from a combinatorial point of view.


invariant partition B such that, for some B ∈ B, the actions of GB on B and ΓB(B)

are permutationally equivalent with respect to an incidence-preserving bijection ρ.

Suppose further that adjacent vertices of Γ have the same second coordinate. Then

there exists a G-invariant set H of hexagons of the graph Γ ∪ Γ′ such that

(a) the edges of each hexagon of H lie in Γ and Γ′ alternatively;

(b) each edge of Γ belongs to a unique hexagon of H, and each edge of Γ′ belongs

to exactly k hexagons of H; and

(c) any two hexagons of H have at most one common edge.

Proof Let “BC”, “DC” be an edge of Γ. Then “BC”ID and “DC”IB. From

(11.3) and our assumption on labels of adjacent vertices, it follows that “BD” is adja-

cent to “CD” and “DB” is adjacent to “CB”. It is easy to see that h“BC”, “DC”

:= (“BC”, “DC”, “CD”, “BD”, “DB”, “CB”, “BC”) is a hexagon, and that its

edges belong to Γ and Γ′ alternatively. (See Figure 9, where the dashed lines repre-

sent edges of Γ′.) Set

H := h“BC”, “DC” : (“BC”, “DC”) ∈ Arc(Γ).

Since both Γ and Γ′ are G-symmetric, H is G-invariant. One can see that

h“BC”, “DC” = h“CD”, “BD” = h“DB”, “CB”,

and that this is the unique hexagon in H containing the edge “BC”, “DC” of Γ.

By Theorem 11.2.2(b), we have Γ[B,D] ∼= k ·K2. When “BC”, “DC” runs over

all the edges of Γ[B,D], we get k hexagons h“BC”, “DC” and these are the only

members of H containing the edge “BD”, “DB” of Γ′. So both (a) and (b) are

true. The validity of (c) follows immediately from the definition of the hexagons of

H. 2


‘‘BC"

‘‘DC" ‘‘BD"

‘‘DB"

‘‘CB"

‘‘CD"

Figure 9 Hexagons in Γ ∪ Γ′

The case where two adjacent vertices of Γ have labels involving four distinct

blocks seems to be much more complicated, even under our additional assumption

that ρ is incidence-preserving. So in the following we concentrate on the extreme

case where Γ[B,C] ∼= k ·K2 is a matching. In this case, we show that there exists a

G-orbit on n-cycles of ΓB, for some even integer n ≥ 4, which determines completely

the adjacency of Γ.


invariant partition B such that, for some B ∈ B, the actions of GB on B and ΓB(B)

are permutationally equivalent with respect to an incidence-preserving bijection ρ.

Suppose further that adjacent vertices of Γ have labels involving four distinct blocks

and that Γ[B,C] ∼= k ·K2 for adjacent blocks B,C of B. Then there exist an even

integer n ≥ 4 and a G-orbit E on n-cycles of ΓB such that the following (a)-(c) hold:

(a) each 2-arc of ΓB is contained in at most one n-cycle of E ;

(b) a 2-arc of ΓB is contained in an n-cycle of E if and only if it lies in arc2(ΓB);

and

(c) two vertices “BC”, “DE” of Γ are adjacent if and only if (C,B,D,E) is a

3-arc of ΓB contained in an n-cycle of E .

Incidence-preserving Case 157

Proof Let (B0, B1, B2) ∈ arc2(ΓB), that is, “B1B0”IB2. Then, since Γ[B1, B2] ∼=

k ·K2 by our assumption, there exists a unique block B3 ∈ ΓB(B2) such that “B2B3”

is the unique vertex in B2 adjacent to “B1B0”. This implies (B3, B2, B1) ∈ arc2(ΓB)

and hence (B1, B2, B3) ∈ arc2(ΓB) by (11.4). Thus “B2B1”IB3 and hence there

exists a unique block B4 ∈ ΓB(B3) such that “B3B4” is the unique vertex in B3

adjacent to “B2B1”. This in turn implies that (B4, B3, B2) ∈ arc2(ΓB) and hence

(B2, B3, B4) ∈ arc2(ΓB). Inductively, suppose that B0, B1, B2, . . . , Bi have been de-

termined for some i ≥ 3 such that (Bj−1, Bj , Bj+1), (Bj+1, Bj, Bj−1) ∈ arc2(ΓB)

for j = 1, 2, . . . , i − 1, and that “BjBj−1” is adjacent to “Bj+1Bj+2” for j =

1, 2, . . . , i − 2. Then in particular “Bi−1Bi−2”IBi and hence there exists a unique

block Bi+1 ∈ ΓB(Bi) such that “BiBi+1” is the unique vertex in Bi adjacent to

“Bi−1Bi−2”. Thus we have (Bi+1, Bi, Bi−1) ∈ arc2(ΓB) and hence (Bi−1, Bi, Bi+1) ∈

arc2(ΓB). Continuing this process, we see that each 2-arc (B0, B1, B2) in arc2(ΓB)

determines a unique sequence B0, B1, B2, . . . , Bi, Bi+1, . . . of blocks of B such that

(Bi−1, Bi, Bi+1), (Bi+1, Bi, Bi−1) ∈ arc2(ΓB) and “BiBi−1” is adjacent to “Bi+1Bi+2”

for each i ≥ 1. Our assumption on labels of adjacent vertices of Γ implies that

any four consecutive blocks in this sequence are pairwise distinct. Since we have

only a finite number of blocks in B, this sequence must contain repeated terms.

Let Bn be the first block in the sequence which coincides with one of the pre-

ceding blocks. Then n ≥ 4 and we claim that Bn must coincide with B0. Sup-

pose to the contrary that Bn = Bℓ for some integer ℓ with ℓ ≥ 1. Then, since

arc2(ΓB) is a G-orbit on Arc2(ΓB) (Theorem 11.3.1(b)), there exists x ∈ G such

that (Bxℓ , B

xℓ+1, B

xℓ+2) = (B0, B1, B2). By the construction above, one can see

that the sequence determined by (Bxℓ , B

xℓ+1, B

xℓ+2) is Bx

ℓ , Bxℓ+1, B

xℓ+2, . . . , B

xℓ+i, . . ..

So by the uniqueness of the sequence determined by (B0, B1, B2) we must have

Bxℓ+i = Bi for each i ≥ 0. In particular, we have Bx

n = Bxℓ+(n−ℓ) = Bn−ℓ. On

the other hand, Bn = Bℓ implies that Bxn = Bx

ℓ = B0. Thus we have Bn−ℓ =

B0, which contradicts the minimality of n. So Bn must coincide with B0 and

we get an n-cycle C(B0, B1, B2) := (B0, B1, B2, . . . , Bn−1, B0) of ΓB. Note that

(B2, B1, B0) ∈ arc2(ΓB) implies that there exists a unique block C ∈ ΓB(B0) such

that “B0C” is the unique vertex in B0 adjacent to “B1B2”. So we have (C,B0, B1) ∈

arc2(ΓB) and, by the construction above, the sequence determined by (C,B0, B1) is

C,B0, B1, B2, . . . , Bi, . . .. Since the first repeated block in this sequence is C, as


shown above, we must have C = Bn−1 and hence (Bn−1, B0, B1), (B1, B0, Bn−1) ∈

arc2(ΓB) and “B0Bn−1” is adjacent to “B1B2”. In a similar way, one can show

that (Bn−2, Bn−1, B0), (B0, Bn−1, Bn−2) ∈ arc2(ΓB) and “Bn−1Bn−2” is adjacent to

“B0B1”. Therefore, reading the subscripts modulo n (here and in the remainder

of the proof), we have (Bi−1, Bi, Bi+1), (Bi+1, Bi, Bi−1) ∈ arc2(ΓB) and “BiBi−1”

is adjacent to “Bi+1Bi+2” for each i ≥ 1. Hence n must be an even integer and,

by definition, all these 2-arcs contained in C(B0, B1, B2) determine the same n-

cycle, namely C(B0, B1, B2). By Theorem 11.3.1(b) any 2-arc in arc2(ΓB) has the

form (Bx0 , B

x1 , B

x2 ) for some x ∈ G, and by definition we have C(Bx

0 , Bx1 , B

x2 ) =

(Bx0 , B

x1 , B

x2 , . . . , B

xn−1, B

x0 ) = (C(B0, B1, B2))

x. This implies that E := C(E,D,B) :

(E,D,B) ∈ arc2(ΓB) is a G-orbit on n-cycles of ΓB. Note that, for a given 2-arc

(E,D,B) of arc2(ΓB), C(E,D,B) is the unique n-cycle in E containing (E,D,B).

So (a) and (b) are true. If “DE”, “BC” are adjacent in Γ, then (E,D,B) ∈ arc2(ΓB)

and by the argument above (E,D,B,C) is a 3-arc contained in C(E,D,B). Con-

versely, from the definition of the n-cycles in E , for each 3-arc (E,D,B,C) contained

in an n-cycle of E , “DE”, “BC” are adjacent in Γ and hence (c) follows. 2

Remark 11.4.1 Suppose Γ, G, B and ρ are as in Example 11.1.1. Suppose further

that Γ almost covers ΓB. Then ρ is incidence-preserving, as mentioned at the begin-

ning of this section, and arc2(ΓB) = Arc2(ΓB) by Theorem 11.3.1(c). So in this case

Theorem 11.4.2 implies that ΓB is a near n-gonal graph with respect to E . Thus,

Theorem 11.4.2 can be taken as a generalization of the first assertion of Theorem

7.0.2(b), and the proof is similar in spirit to that of the “only if” part of Theorem

7.3.1.

11.5 Reconstruction of Γ, and 3-arc graphs again

By using the labelling technique established in Section 11.2, we now prove that any

G-symmetric graph Γ satisfying [P] can be reconstructed from the quotient graph ΓB

and the action of G on B, namely Γ is isomorphic to a 3-arc graph of ΓB with respect

to a certain self-paired G-orbit on 3-arcs of ΓB. Conversely, we prove that, for any

G-symmetric graph Σ and any self-paired G-orbit ∆ on Arc3(Σ), the 3-arc graph

Ξ(Σ,∆) satisfies the condition [P]. The proof of the following theorem is essentially

Reconstruction 159

the same as the proofs of Theorems 5.2.1 and 5.2.2.


G-invariant partition B such that the actions of GB on B and ΓB(B) are permu-

tationally equivalent, so the vertices of Γ are labelled by ordered pairs of adjacent

blocks of ΓB. Then Γ ∼= Ξ(ΓB,∆) holds for ∆ the (self-paired) G-orbit on Arc3(ΓB)

containing the 3-arc (C,B,D,E), where (“BC”, “DE”) is an arc of Γ.

Conversely, for any G-symmetric graph Σ and any self-paired G-orbit ∆ on

Arc3(Σ), the graph Γ := Ξ(Σ,∆), group G and partition B := B(Σ) satisfy all

the conditions above. Moreover, we have ΓB∼= Σ.

Proof Let Γ, G and B be as in the first part of the theorem. Let (“BC”, “DE”)

be a fixed arc of Γ. Then by Theorem 11.2.1(d), (C,B,D,E) is a 3-arc of ΓB. Let

∆ be the G-orbit on Arc3(ΓB) containing (C,B,D,E). Since Γ is G-symmetric,

there exists x ∈ G such that (“BC”, “DE”)x = (“DE”, “BC”). So (E,D,B,C) =

(C,B,D,E)x ∈ ∆ by (11.1), and hence ∆ is self-paired. Again by the G-symmetry

of Γ and (11.1), we have: (C1, B1, D1, E1) ∈ ∆ ⇔ there exists x ∈ G such that

(C1, B1, D1, E1) = (C,B,D,E)x ⇔ there exists x ∈ G such that (“B1C1”, “D1E1”) =

(“BC”, “DE”)x ⇔ (“B1C1”, “D1E1”) ∈ Arc(Γ). Therefore, the mapping “B1C1” 7→

(B1, C1), for “B1C1” ∈ V (Γ), establishes a graph isomorphism from Γ to Ξ(ΓB,∆).

Now suppose Σ is a G-symmetric graph and ∆ is a self-paired G-orbit on Arc3(Σ),

and let (τ, σ, σ′, τ ′) ∈ ∆. Then from Lemma 5.2.1, Γ := Ξ(Σ,∆) is a G-symmetric

graph with B := B(Σ) a nontrivial G-invariant partition of V (Γ), where B(Σ) :=

B(α) : α ∈ V (Σ) is as defined in Section 5.2. IfB(α) andB(α′) are adjacent blocks

of B, then there exist (α, β) ∈ B(α) and (α′, β ′) ∈ B(α′) such that (α, β), (α′, β ′)

are adjacent in Γ, and hence (β, α, α′, β ′) ∈ ∆. In particular, we have (α, α′) ∈

Arc(Σ). Conversely, suppose (α, α′) ∈ Arc(Σ). Then since Σ is G-symmetric there

exists g ∈ G such that (σ, σ′)g = (α, α′). Setting τ g = β and (τ ′)g = β ′, then

(β, α, α′, β ′) = (τ, σ, σ′, τ ′)g ∈ ∆. So (α, β) ∈ B(α) is adjacent to (α′, β ′) ∈ B(α′)

in Γ and hence B(α) and B(α′) are adjacent blocks of B. Thus, α 7→ B(α) defines

an isomorphism from Σ to ΓB. From Lemma 5.2.1(d), the actions of GB(σ) on B(σ)

and Σ(σ) are permutationally equivalent with respect to the bijection (σ, σ′) 7→ σ′.

So the actions of GB(σ) on B(σ) and ΓB(B(σ)) are permutationally equivalent with

respect to the bijection ρ : (σ, σ′) 7→ B(σ′), for (σ, σ′) ∈ B(σ). 2


Remark 11.5.1 (a) Theorem 11.5.1 is a counterpart of the second part of Theorem

5.2.3, where ΓB and Σ are assumed to be (G, 2)-arc transitive.

(b) From Theorem 11.4.2(c) and the proof above one can see that, under the

assumptions of Theorem 11.4.2, the self-paired G-orbit ∆ on Arc3(ΓB) such that

Ξ(ΓB,∆) ∼= Γ is precisely the set of all 3-arcs of ΓB contained in some n-cycle of E .

Conversely, if, for a G-symmetric graph Σ, there exist an even integer n ≥ 4

and a G-orbit E on n-cycles of Σ such that each 2-arc of Σ is contained in at most

one n-cycle of E , and that the set of 2-arcs of Σ contained in some n-cycle of E is a

G-orbit on Arc2(Σ), then one can check that the following (i)-(iii) hold:

(i) the set ∆ of 3-arcs of Σ contained in some n-cycle of E is a self-paired G-orbit

on Arc3(Σ), and thus Γ := Ξ(Σ,∆) is well-defined;

(ii) for B := B(Σ), the bijection ρ from B(σ) to ΓB(B(σ)) defined at the end of

the proof of Theorem 11.5.1 is incidence-preserving; and

(iii) Γ[B(σ), B(σ′)] ∼= k ·K2 for adjacent blocks B(σ), B(σ′) of B.

These results together give the inverse of Theorem 11.4.2 and generalize the second

assertion in Theorem 7.0.2(b).

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Appendix A

The Chinese originals of thequotes

The quotes used in the thesis are translated from some Chinese classics 1. Listed

below are the Chinese originals and the chapters of the thesis where these quotes

are used.

1We referenced the translations of these classics in the following books: 1. A Source Bookin Chinese Philosophy, translated and compiled by Wing-Tsit Chan, Princeton University Press,Princeton, NJ, 1963; 2. The Analects of Confucius, translated and annotated by Arthur Waley,George Allen & Unwin Ltd, London, 1938; 3. Confucius: The Great Digest and the UnwobblingPivot, translated and commented by Ezra Pound, Peter Owen, London, 1952.

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